Both the 9x23 Win & 10mm Auto will give up roughly 75fps going from a 5" to a 4" barrel, so I would extrapolate that to roughly 100 to 110 fps if you go to a 3.5" barrel. The standard 9x23 load is a 125g bullet at 1400fps from a 5" barrel, yielding approximately 1300fps from a 3.5" barrel. For the 10mm lets use a 180g bullet at 1200fps from a 5" barrel stepping down to 1100fps in a 3.5" barrel. From a terminal ballistics standpoint that is still going to give you substantially above 9mm ballistics (from a 4"+ barrel) from the 9x23 Win out of a 3.5" tube. Comparatively the 10mm out of the shorter barrel will yield a similar comparison to the .40 S&W with the 3.5" 10mm still surpassing the .40 out of a "service length" barrel. Of course it is more complicated than just velocity: sectional density, penetration, etc but you will lose similar velocity from both rounds and both rounds will still beat their smaller counterparts even with the abbreviated barrels.
I myself love the 10mm & the 9x23 Winchester, have multiple 10mm handguns and will have multiple 9x23 Win guns before I am done (just one ATM). I will say that anyone that feels 10mm is an "uncommon" caliber has never shopped for 9x23 Winchester. I reload for all the calibers I shoot so to me that is a moot point but may factor into your decision. The obvious answer is to build one of each

but that doesn't answer your question.
Function of each in the 1911 or in this case the 2011 platform generally requires a little more care in the building as they are non-standard calibers for the platform. Due to heavier bullet mass the 10mm will have more recoil than the 9x23 which to me is irrelevant in the shooting of the guns but needs to be factored in to the building of said guns. Being that you are using the 2011 platform I will assume you are using ramped barrels? Not a must IMO but I know a lot of the 2011 pistols come with the ramp cut done. Both the 10mm & 9x23 cases are strong enough that I don't feel ramped barrels are a necessity but I have also not done any 2011 builds with an unramped barrel (for no reason other than the 2011 frames I have utilized were all pre-cut for ramped barrels).
With both rounds being smaller in diameter than the .45 ACP they are presented higher in the magazine so feeding becomes more of a straight line affair, with some 9x23 magazines presenting the round so it almost feeds directly into the chamber. Magazine capacity is also greater with the 9x23 by a fair amount.
As I said before I love both rounds and promote them whenever I can but I've never had to pit them against one another. Some of it comes down to your philosophy on terminal ballistics insofar as are you a light & fast or heavy & slow guy? Of course the 10mm isn't really "slow" but in this case it is slower than the comparison round. Power of the rounds, the slide velocity and their recoil will make the 10mm a little more sensitive to the building and adjusting of the timing & recoil system versus the 9x23.
So on paper I guess I've made the argument for the 9x23 (although inadvertently) but again my true answer would be build them both

It is certainly up to you, but if you build a 10mm personally I would abandon the .40S&W switch barrel idea as the shorter rounds will more than likely require special tuned magazines and with the 10mm you can load light to replicate the 40 short & wimpy. In 9x23 Win I would still consider the 9mm barrel just because the price & availability of 9x23 brass versus standard 9x19mm.