Is it possible to represent a regular expression with bounded captures using a DFA and O(1) additional processing?
I have no direct answer, but some additional remarks regarding on the topicality of my paper from this year's STACS which Raphael  mentioned in his answer. (Due to space reasons, I opted for a new answer, instead of a comment.)

As far as I understand the question, the classes of regular expressions with $\leq k$ captures corresponded to the classes RegEx(k) of extended regular expressions with at most $k$ variables in my paper. Among the results in the paper, the ones that are probably most relevant to the question are that, even with one variable (the class RegEx(1)),

1) there is no effective way of converting expressions from RegEx(1) to "proper" regular expressions, even if the problem is restricted to those expressions that describe regular languages,

2) even if one is able to convert these expressions with one variable to proper regular expressions (or any other description mechanism for the class of regular languages), the resulting blowup in size is not bounded by any computable function,

3) given an expression $\alpha\in RegEx(1)$, universality (the question whether $L(\alpha)=\Sigma^*$) is not decidable.

This does not directly answer the question, but imposes some limits on possible conversion procedures. For example, if the universality problem for the class of resulting DFAs with extra storage is decidable, such a conversion procedure cannot exist. (On the other hand, although the automata model that is described in the question appears to be underspecified regarding the storage and storage operations, I am under the impression that this problem should be undecidable for the resulting class of automata.)

Among the few papers that deal with regular expressions with variables/backreferences/captures, I know of only two other works that consider something in the direction of limiting the number of variables: Aho's chapter on string matching in the Handbook of TCS (Vol. A), which contains a sketch of a proof that extended regular expressions with at most $k$ backreferences can be matched in time $O(n^{2k})$, and the paper by Reidenbach and Schmid from CIAA 2010, which gives a polynomial time matching algorithm for a larger class of extended regular expressions.