Basement – How to dress up a boring drop ceiling

Per the wording of that code, you can't be adding anything to the floor or the ceiling as you'll be under the 7' minimum. You could consider a super-thin flooring (stain the concrete? Linoleum?) and then, as you suggest, put the sheetrock between the joists (though that sounds like a finishing nightmare).

Alternatively, raise the foundation (likely cost prohibitive).

All that said, I echo B Mitch...what is the concern of not meeting the code? The main drawback would be that you couldn't advertise it as finished square footage when you sell.

The first thing I'd do, though, is have an inspector come out and ask what the odds would be in getting a 1" variance or so. They may count it as 'close enough' and let you get by.

You may be able to hang birch- or maple-faced plywood. You can screw it to joists and then unscrew a panel when you need to get in to change stuff.

My neighbor used this as a floor, with a urethane coating, but you may be able to skip the finish.

I haven't tried this approach myself. YMMV.