I myself don't know if any Aristotle interpreters have responded to this or where best to find a critique, but I wonder if we could understand Rawls in a way that agrees with you when you say:
Isn't Aristotle's view that our natural capacities are exercised not just through any natural activity like sight but through rationally directed virtuous activity in particular, where this requires a particular sort of education and political structure.
Though it might be possible to conceive of a rationally directed virtuous activity that does not involve a natural activity, insofar as we are not merely rational minds but are also nutritive organisms (like plants) and perceptive/active organisms (like non-human animals) it seems to me that all rationally directed virtuous activity involves natural activity (if by that you mean our senses). Aristotle does believe that human excellence involves what is uniquely human (our rational nature 1097b-1098a (all translations are David Ross')). But I don't believe Aristotle thinks that there is no excellence of non-rational things, as when he says "for it is by the excellence of the eye that we see well." (1106a). Nor does he seem to think that human excellence does not involve non-rational natural activities/capacities, as when he says:
Now the activity of the practical virtues is exhibited in political or military affairs... (1177b)
since practical virtues require sense. Even for those senses that we might seem to share in whole with non-human animals Aristotle thinks humans have a different relationship to them, as when he writes:
And so too is it with objects of hearing; no one calls those who
delight extravagantly in music or acting self-indulgent, nor those
who do so as they ought temperate... Nor is there in animals other than man any pleasure connected with these senses, except incidentally. For dogs do not delight in the scent of hares... (1118a)
Here Aristotle characterizes hearing, a capacity shared with non-human animals in a way that rationality is not, as providing a pleasure to humans by virtue of them not being non-human animals, and this I think contributes to the view that it is not rationality apart and alone that Aristotle thinks good or virtuous but that our senses also are at least necessary for many forms of virtue (running courageously into virtue) and can contribute to pleasure/enjoyment.
Aristotle does not identify the life of the philosopher as the only virtuous life. He does think it is the happiest life, but this arises out of considerations of the role of external goods in the actualization of other virtues ("the brave man will need power if he is to accomplish any of the acts that correspond to his virtue, and the temperate man will need opportunity" (1178a)) and an argument for contemplation being more favored by the gods (1179a).
In the context of a discussion of pleasure and the ideal relationship to it Aristotle writes:
The temperate man occupies a middle position with regard to these objects...
the things that, being pleasant, make for health or for
good condition, he will desire moderately and as he should, and also
other pleasant things if they are not hindrances to these ends, or
contrary to what is noble, or beyond his means. (1119a)
From that passage it seems Aristotle thinks pleasure intrinsically valuable such that its cultivation is not problematic to pursue unless doing so would be unhealthy, or ignoble, or imprudent. So, as regards pleasure I think Rawls' view of enjoyment is not prima facie in tension with Aristotle, unless Rawls' also believes some things about enjoyment I will talk about below.
In relation to the example of chess and checkers and the notion of complexity, some other passages from the Nicomachean Ethics are maybe relevant, as when Aristotle writes:
Now if you take away from a living being action, and still more production, what is left but contemplation? Therefore the activity of god, which surpasses all others in blessedness, must be contemplative; and of human activities, therefore, that which is most akin to this must be most of the nature of happiness. (1178b)
I wonder if we can think of Rawls' 'Aristotelian principle' as an object-level claim about what greater and lesser degrees of contemplation are. It seems to me like a plausible interpretation of Aristotle's claims about the goodness of contemplation and the greater goodness of greater degrees of contemplation as capable of being mapped onto the kind of complexity scale along which chess is more complex than checkers. From the perspective of modern animal psychology, we moderns might think that some non-human animals might engage in something akin to the contemplation humans do, e.g. dolphins or elephants. Even if we do think this, though, I think an Aristotelian would maintain that the contemplative capacity of humans exceeds that of non-human animals. Likewise, the exercise of our contemplative capacity seems to exist on a scale from more to less contemplative. Aristotle I think believes in there being more and less developed capacities and the objects those capacities are exercised on:
... while there is pleasure in respect of any sense, and in respect of thought and contemplation no less, the most complete is pleasant, and that of a well-conditioned organ in relation to the worthiest of its objects is the most complete... (1174b)
If by 'enjoyment' Rawls' means nothing other than positively valenced qualia I do think his principle would be too hedonistic to be entirely Aristotilean; however, we could understand 'enjoyment' in a wider sense that encompasses uses like 'I enjoyed Titus Andronicus (1999), though I hard time stomaching some of the scenes' or 'I enjoyed running a marathon, painful though it was.' In that wider sense I do think Aristotle claims we do enjoy "the exercise of our realized capacities," as Rawls claims. Aristotle also thinks we should orient ourselves in particular ways and cultivate particular capacities, but Rawls' addendum of "and this enjoyment increases the more the capacity is realized, or the greater its complexity" I think carries a similar normative weight if Rawls thinks that enjoyment has normative force. I'm not familiar enough with Rawls to say that he does think that, but if he does then I think Rawls might not be vulnerable to the worries you raise above. I might be mistaking the concern though, or misunderstanding the potential hedonism of Rawls' principle.
Aristotle also writes:
The most valuable possession is that which is worth most, e.g. gold, but the
most valuable work of art is that which is great and beautiful (for
the contemplation of such a work inspires admiration, and so does
magnificence); and a work has an excellence — namely, magnificence — which
involves magnitude. (1122b)
Though he does not expand on the notion of 'magnitude' in relation to the excellence of a work of art in the Nicomachean Ethics, I think here, as in passages above, is evidence of a notion of 'scale' of worth that I think Rawls' example of chess and checkers is plausibly mapped onto.
Misc.
Related to the second part of Rawls' principle, Aristotle writes:
...for a good man qua good delights in virtuous actions and is vexed at vicious ones, as a musical man enjoys beautiful tunes but is pained at bad ones. (1170a)
I'm guessing (But only guessing!) that the first part of Rawls' principle is the one that moreso seems in tension with Aristotle than the second part, so I focused on that above, but if I write aught of worth and if the second part poses more trouble I can write more of it!
(Also, perhaps less relevant: I agree Aristotle does believe in the usefulness of a social order for promoting individual virtue (1180a) though I do not think participation in a social order is necessary for individual virtue (not that you said that) ("Now if [the good man] were a solitary, life would be hard for him; for by oneself it is not easy to be continuously active; but with others and towards others it is easier." (1170a).)
