Greg - I'm sorry about the diatribe for backlash - and I am quite sure you understand it. I just get carried away sometimes.
Do I understand that this circular table, using the angular axis, will move to a new position, and then stand still whilst the other axis go to work on it.
If so - I know this sounds crude - but I would put a parking brake on it.
There are several options - a belt round the table that can wind on like a garrot, an electric disc brake that might be better because you could control it through software, or my son ( who weighs 22 stone and loves going down mountains on his bike) puts his faith in bicycle brakes - the latest disc brakes are awsome in their power - I now use them on miniature railway engines. They are compact and easy to fit and have hydraulic or cable operation. Or, in the picture you have posted, a drilled and taped hole for a screw onto the main shaft of the gearbox.