Yes a Tiagra, or any Shimano road cassette, will work fine with your MTB components. I'm not surprised you never use your low gears and an 11x23 9-speed cassette should give you an excellent choice of gears with a low gear way beyond anything Florida requires.

Years ago I had a bike I kept at a friend's house in Orlando that was geared 50/40/30 chainrings and had a 13x21 7-speed cassette. I never needed the 30T chainring and that cassette gave me a "straight block" (1 tooth cog differences) from 13 through 17T. Even "Sugar Loaf Mountain Road" never needed the lowest gear and there were no downhills requiring a tall high gear.