Re: To bow or not to bow
The formalization of rei is part of what we do but does not define it. Like you pointed out - Aikikai Honbu makes the accommodation and generally a bow can be anything from a simple nod to prostration, the latter which can cause concern beyond Islam.
In a similar situation I only asked that the practice of the others not be disrupted and a certain times some indication be made. It ended up being a verbal expression (which everyone was doing) and a slight nod. Years ago I had a different faith orthodox make things very difficult but in that case there was more than faith at work I think. Luckily in that case it only lasted a couple of lessons.