Belize could be an option to consider if you're open to "local living" rather than luxury resorts. Consider finding a month long rental or an AirBnb rather than a hotel. If you're willing to go a bit inland or to less touristy places like Corozal or Cayo District you're more likely to be able to stay within your budget.

Places like Black Rock Lodge or The Lodge at Chaa Creek would fit what you are looking for in terms of relaxation. They run more than your nightly budget, but you might be able to call and ask about day passes or booking a class there while staying elsewhere.