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Good controller!
Without worrying any more about drain, why don't you look for a 30-40 Ah battery. That should do what you want - one 4" strap and one 8" strap, not running at 100% all the time.
OK, I'd better try to estimate. A 4" DewNot strap draws under 0.2A and an 8" draws under 0.4 A - lets say 0.6A. Let's add a bit for the controller to dissipate - 0.2A should be plenty - total 0.8A. OK, between friends, let's make that 1A to be generous. For 8 hours that is about 8Ahr. Let's double that so you don't discharge the batter too far - say 16Ahr. So even a 20Ahr battery should be plenty. (But will you add more straps in future??)
If you have a second hand option in batteries, that is worth pursuing - $30-40 is worth a try.
But I'd be looking looking for backup batteries for electrical equipment. They come on the market all the time and have probably spent their entire life on trickle charge and never were discharged. I'd worry that wheelchair batteries may have been discharged too low a few times.
Any used battery, you want to see tested - not just voltage, but voltage under load. I picked up one for someone the other day and the second-hand supplier put a big Arlec meter on it to show me firstly voltage unloaded and then that it held a respectable voltage under a reasonable load.
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