How to hang exhaust fan with limited amount of space?

Google "remote exhaust fan" and see if that will work. The fan portion doesn't need to be in the bathroom ceiling.

Drop ceilings are a special case. With small tiles I make a flange if it is rated for grid support.

Most are not grid rated for larger tiles 2-4 and require standard tie wire (galvanized wire attached to the structure not conduit) if in a seismic area like I am in grid support is usually prohibited. I have never tried chain but don’t see why it could not be used (chain may be noisy compared to tie wire).

You don't hang an exhaust fan because you have a drop ceiling. You are making this WAYYYY too hard. (Also a drop ceiling means you have more space, not a limited amount)

Your fan may be able to be mounted - they are all different shapes and sizes - low enough to even up with drop ceiling and it it can't be it just takes a few pieces of scrap lumber to add lower supports for your fan. But you would need to supply more info for that.

Also I can't fathom why you would want a drop ceiling in your basement bathroom. It is fine if the rest of the basement will be drop ceiling but drywall the bathroom and hit it with an oil based primer on all walls. You drop ceiling there then a good percentage of the moisture supposed to go to fan will end up in your ceiling, you will have rust and mold issues if that bathroom is used a lot.