I'm in small room myself, trying to make the best use of space.
Consider a Japanese futon. I've been using one for the last five months and I love it. It is...maximally firm. Personally I like the "primal sleeping" aspect, and it helps a bit with my sciatica.
I'm using: https://www.amazon.com/Rolling-Bed-JFM-36080-Black-60inch-Mattresses/dp/B009EEMH3Q
Lately I've been considering a bit more padding.. possibly going with a legit tatami mat, otherwise a firm 1" foam pad.
The futon folds up quickly and fits in the closet. Plenty of room for a narrow bench sofa to use during the day.
Depending on the issue, sometimes there are cheap fixes. We've placed plywood sheets directly under our mattress if it had too much "give". We've also just given up on the mattress concept before and gotten much cheaper shikibuton-style pads directly on a floor or plywood sheet over the slats that took some initial getting used to but felt much better than a bad mattress after the initial adjustment.
I've had a shikibuton for just over a year and I don't think that I will switch back to a traditional bed any time soon.
I just bought a jlife shikibuton and it was the best thing I could have done. I have really hard, terrible carpet so I bought an area rug off amazon and it helps. There is a break in period. Your body is so used to a traditional bed that it's gonna take some time to adjust.
What my husband and I did, was buy this bed with a tri fold mattress and slept on them for the first year and then last month we made the switch to the jlife one that is more authentic.
If you're on the thinner side, you should be fine with just the D&D Bed but since my husband and I are larger, we needed the extra padding and had to buy the tri fold pad. With the Jlife 3 inch futon that we bought, we are good with just that.
I never had issue with the fitted sheets. I even had the waterproof mattress protector and a fitted sheet on there and it was fine :)
Ive had less overall back pain since switching to a shikibuton.
https://www.amazon.com/dp/B009EEMH3Q/ref=cm_sw_r_cp_apa_glt_fabc_WXYDHSH6NZXSPERN4XS6
I've been using one of these for about three years and love it.