Why would an approach be Cat A NA?
Original answer: The speed of Category A aircraft is too low to execute the missed approach.  For this particular airport, it's basically a hole in the ground with steep mountains on all sides. To go missed, you gotta get up in a hurry, and the TERPS data probably indicates a minimum speed is needed.  



Edit:  Despite the downvotes and comments stating that my answer was "unlikely", I'm sticking to my guns. 

To make sure that I knew what I was talking about, I contacted the FAA's Flight Procedures Standards Branch in OK City and eventually was put in contact with someone who was able to pull up the documentation on this approach. 

Listed in the comments of the document he was looking at was this: "Ref 8260.52 CAT A -- final approach speed is too slow for RNP missed approach segment length for obstacle penetration."

Translation: whoever put this approach together did the math that is required, and the formula result was that speeds needed for obstacle clearance are not sufficient when using CAT A speeds. 

Note 1: some math and formulas required to build an IAP missed approach procedure can be found in Chapter 4 of 8260.52. (https://www.faa.gov/documentLibrary/media/Order/ND/8260_52.pdf)

Note 2: 8260.52 has since been superseded by 8260.58, which itself has undergone an update and the most current document is 8260.58A.



Gory technical details (or some insight as to why this oddity is the way it is):

The thing that makes this approach special is that not only is it a RNP AR approach, the missed approach requires abnormally precise navigation as well (i.e. RNP values &lt;1.0).  The length of this span of the missed approach (from the MAP to TIPOC) is bounded by the results of formula 4-3-8 in 8260.58A, which is dependent on the true airspeed of the slowest aircraft of the approach.

DMASRNP is the max distance allowed for a given true airspeed (category) to return to RNP 1.0 on missed approach. The longer the distance to return to RNP 1.0, the faster the plane needs to be.

In the case of the KGUC RNAV (RNP) 6 -- the MAP to TIPOC span is 6.7NM along track, which is longer than the formula's results no matter which of the three possible RNPs are applied if you are using a category A airspeed (these results are applied to the whole span as per figure 4-3-5).

Why this formula is needed, and applied to the whole span for that matter, is a mystery, though -- 8260.58A gives no design rationale for this limitation, or why it is pegged to airspeed.  A rationale is that navigation errors are cumulative though -- the slower you're flying, the more time you have to drift off course before you make it to the next "gate" where a higher RNP prevails.

Other approaches (even in similar terrain and with seemingly similar conditions charted) may not have this peculiar limitation, though -- the KRIL RNAV (RNP) Z 26 approach, for instance, also requires RNP &lt; 1.0 on the missed approach, but for a shorter distance -- a mere 3.5NM from the MAP to TEROE, whereas the limit distance for Category A given the RNPs used in the missed approach there is 4NM.

Moral of the story: pay attention to your approach plates!