I would apply for the permits and find out what the inspector has to say.
And, just because there is a legitimate ceiling height factor doesn't mean you can't use spaces in your home for whatever you like. There are many 1 1/2 story homes that have upper level spaces with rather low ceiling height, but many home owners use the spaces for bedrooms or offices or whatever.
It would come in to play when you go to sell. Make sure you disclose that the those spaces might be considered "non-conforming". That usually spells it out to those who are inquiring!!
Raising the ceiling can be an exceptionally expensive endeavor. The cost would likely not be worth the addtional space that you will receive. I would simply submit your request for permit and let the city tell you that you cannot have the space down there. Even then, you might be able to fight the ruling through the planning and zoning board (seeking a varianace) or the city council.
There are ways to work around the current joist system and drop your soffits where needed to maximize headroom. I would definitely look into a system that is attached directly to the underside of the joist and not a drop or hanging ceiling sort of system.
Coldwell Banker Burnet
licesned MN Real Estate Broker
departments. Good luck! Liz, 612-986-4105