ChrisAttebery has it dead right. Keep the 10" props in your "emergency spares" box. They just don't give everything that there is to be had from a BB .46 like a TT Pro -- especially with a "draggy" plane like a trainer.

I've flown many Avistars with OS FXs & TT Pros -- The 11 & 12" props are where you want to be. An 11-5 is good -- unlimited vertical. An 11-6 will give you a bit more speed (not that you need it) without sacrificing vertical, but will raise the landing speed a little. An 11-4 works OK if you stay away from full throttle in level flight, but it will pull an Avistar straight up 'till it's gone, and it lowers the landing speed. A 12-4 is exactly right in every way, except ground clearance -- & you can get that if you convert to a tail dragger.

While APC props really do give better performance than MAS, they break much more easily. The TT will have so much go in the Avistar that you won't need the little bit extra available from APC.

Regarding balance & the battery -- move it aft as far as necessary -- don't add useless weight like lead to balance the plane.