Is there any research of spelling errors from a dual-process theory perspective? Checking the rather comprehensive but somewhat dated 2008 review of  Evans on dual-process theories, I didn't see anything about [mis]spelling. 

But it seems to me an obvious thing to investigate from a dual-process perspective, because (in theory) the more elaborate rules you have to apply the more mistakes you're likely to make under time pressure.

So has anything like this been attempted?

Note that this is a little more sophisticated than simply asking if writing (never mind typing) faster results in more errors. You'd have to discern what's the likely cause of the errors, i.e. under time pressure, do you make more errors of the kind that require more complex "graphotactic" rules?
Well, it turns out there is some related research, but there are complications because spelling &amp; writing is harder to investigate (compared to syllogisms, for instance)... 


  Spelling relies on two sets of processes, depending on word
  frequency (for a review, see Tainturier and Rapp, 2001; Bonin,
  2003). For frequent words, the letter sequence is directly retrieved
  from the orthographic lexicon via the lexical route. For rare
  or unknown words, the letter sequence is arrived at indirectly,
  through a process of phoneme–grapheme conversion (assembled
  route). The resulting orthographic representation is stored in a
  graphemic buffer (Caramazza et al., 1987; Hillis and Caramazza,
  1989) until its graphomotor execution (i.e., initiation and implementation of motor programs and neuromuscular execution).
  Although the orthographic representation obviously needs to be
  retrieved or calculated prior to execution (i.e., during the writing
  latency; Bonin et al., 2002), Delattre et al. (2006) have shown that
  the orthographic processing of irregular and infrequent words is
  still generally incomplete at this stage, and therefore, continues
  during the graphomotor execution, thus, increasing handwriting duration. According to the authors, these results can be
  interpreted within the dual-route model. When a conflict arises
  between the outputs of the lexical and assembled spelling routes,
  it takes time to resolve, and the central processing of this conflict
  then cascades onto the graphomotor execution.


So yes, there is a posited "dual-route" model for spelling/writing, and yes, it's based on timing experiments... although these did not involve time pressure (or counting spelling errors by type), but rather is based on timing the writing of well-known vs obscure words. Whether this "dual route" neatly matches more conventional dual-process accounts of cognitions is a bit unclear to me.

The complications come from the fact that spelling and writing form a cascading process:


  Three factors seem to determine the temporal course of the
  central processing of spelling. The first one concerns the word’s
  lexical characteristics. The less frequent and/or regular the word,
  the longer the central processing takes and the more likely it is to
  be cascaded. The second factor concerns time pressure. Until now,
  the cascading phenomenon has only been observed for the production of successive words under time pressure (Delattre et al.,
  2006; Lambert et al., 2011; Maggio et al., 2012). Producing a single
  word without any time constraint should minimize cascading, by
  allowing the spelling process to be completed during the pause
  (i.e., latency) preceding execution. Third, according to capacity theory, the ability to cascade processing depends heavily on
  the writer’s development and expertise (Berninger and Swanson,
  1994; Alamargot et al., 2010). Because handwriting is not yet fully
  automatized in younger writers, especially those still in primary
  school (Bourdin and Fayol, 2000; Olive et al., 2009), they have to
  prevent overloading by segmenting and sequentializing the higher
  processes and are therefore, unable to engage in parallel, or cascaded, processing (see Olive and Kellogg, 2002; Alamargot et al.,
  2007).


Interestingly Kahaneman was also involved in formulating capacity theory. And it seems related to models of divided attention, which he also was involved with.

Anyway, if this is correct, an interesting conclusion is that if you have produce the spelling under time pressure and in a medium that you have little experience with (say you seldom handwrite, or you seldom type) then you are more likely to make spelling errors in that medium not simply because of mechanical issues, but also because it puts a sort of cognitive/time pressure even on processes that you have trained in (spelling). 

The introduction of the Delattre paper also has some more interesting observations related to the complexity of the rules, particularly with respect to the how phonemic the orthography is and some empirical support from of specific forms of dysgraphia:


  The dual-route model of spelling production (e.g., Ellis, 1982)
  proposes that two processing systems operate in parallel: a lexical
  route that retrieves spellings of known words from a memory store
  of word-specific knowledge and a nonlexical (or assembled) route
  that generates spellings using a process of sublexical sound-to spelling
  conversion. The assembled spelling route would be efficient
  in languages whose orthographies have predictable or consistent
  orthographic-to-phonological correspondences (such as
  Turkish, Italian, and Japanese kana) but would be considerably
  less effective for English and French, whose orthographies are
  characterized by highly inconsistent relationships (e.g., the vowel
  /i:/ is spelled in many different ways in English words, as in eel,
  tea, theme, thief, Keith, people, me, key, quay, ski, etc.). There are
  many irregular and some almost arbitrarily spelled words in English
  (e.g., pint, yacht) and French (e.g., fraise, monsieur). The
  lexical route would work for all known words (irrespective of
  regularity) but could not provide spellings for new words or
  nonwords. The assembled route would work for nonwords but
  would often produce phonologically plausible errors (PPEs), particularly
  to irregular words, such as yacht (YOT) and monsieur
  (MESSIEU).
  
  Evidence consistent with the dual-route model comes primarily
  from studies of the spelling performance of neuropsychological
  patients with acquired central dysgraphia (for reviews see Barry,
  1994; Tainturier &amp; Rapp, 2000). The separation of the dual routes
  is supported by the double dissociation between surface dysgraphia
  (e.g., Beauvois &amp; De´rouesne´, 1981; Hatfield &amp; Patterson,
  1983) and phonological dysgraphia (e.g., Shallice, 1981). Surfacedysgraphic
  people accurately spell more regular than irregular
  words (for which they make many PPEs) and are interpreted as
  having an impaired lexical route that forces overreliance on their
  preserved assembled route. Phonological-dysgraphic people have a
  marked impairment of nonword spelling but have relatively preserved
  word spelling and are interpreted as having an impaired
  assembled route but an intact lexical route.
  
  Although the lexical and assembled spelling routes are proposed
  to be separate, there are at least two lines of evidence to suggest
  that they interact at some level in normal spellers. First, observation
  of spelling errors in free writing (e.g., Ellis, 1979; Hotopf,
  1980) suggests that assembled spelling plays some role in writing;
  people sometimes make PPEs (e.g., Ver-knickers ayfasia) and may
  produce different alternative spellings on different occasions
  (which suggests that not all errors reflect inaccurate spelling
  knowledge). Second, there is experimental evidence showing lexical
  influence on assembled spelling; there are lexical priming
  effects on nonword spelling (e.g., Barry &amp; Seymour, 1988; Campbell,
  1983; Perry, 2003). For example, the nonword /vi:m/ is more
  likely to be spelled as VEAM after hearing the word team, as
  VEEM after the word deem, and as VEME after the word theme.
  
  Dual-route models of spelling have been expressed in symbolic
  terms (e.g., Barry, 1994) but also as more interactive (Rapp,
  Epstein, &amp; Tainturier, 2002) and connectionist (Houghton &amp; Zorzi,
  2003) models. Although single-route models of spelling have been
  proposed (e.g., Brown &amp; Loosemore, 1994; Bullinaria, 1997;
  Olson &amp; Caramazza, 1994), Houghton and Zorzi (2003) have
  argued that these are currently unable to account for the full range
  of empirical data.


I quoted all this here because the Wikipedia article on spelling has basically zero psychology or neurology information.