Normally I don’t agree with “screw the permits” but yeah as a “trade professional” who has done plenty of work in Westchester (Tarrytown, Larchmont, Irvington, elsewhere)… don’t get a permit. It’s a bigger headache by far than anywhere else I’ve ever worked.

Throw some PT 2x4 flat 16” on center in the floor, fill the space between with 1.5” thick foam insulation sheets, cover with 5 mil plastic sheeting to keep moisture down, and cover with 2 layers of subfloor, seams offset. It’s a classic garage workshop method and it’s worked well for me. I don’t deadlift in the shop, but it’s good on my knees and it doesn’t bow under my 3,000 pound table saw. It’ll handle deadlifting fine, especially if you had a horse stall mat in an area.

Put Rock wool insulation in the walls so you don’t have to worry about moisture issues, and sheetrock. Add an electric heater and some fans, you don’t need to spend on a heat pump or PTAC unit, as others have said - it’s a gym. Keep it warm enough and use fans, breeze, and the big open garage door to cool off in summer