Hi. As far as a rental car, they actually are pretty cheap in SXM, and would allow you to tool around on your own and explore. They do have a couple of souvenir shops at the airport, if you're not real picky about what you want.
I think your biggest issue is going to be wanting to get an early check in at any of these hotels on a Saturday. Many of the hotels on SXM are also timeshare resorts, which mostly have a Saturday turnaround day, so checking in early on a Saturday is probably going to be very difficult. A suggestion for that would be to book the Friday night also. Or take a cab to the hotel OR rent a car and go drop off your luggage at the hotel and have them hold it for you at the front desk until you can actually check in, and go somewhere else for the day. IF YOU DECIDE TO RENT A CAR, DO NOT UNDER ANY CIRCUMSTANCES EVER LEAVE YOUR LUGGAGE IN THE CAR WHEN YOU ARE NOT IN THE CAR. ANYWHERE, AT ANY TIME, FOR ANY AMOUNT OF TIME. If you have a car, you can go anywhere you want for the day, until your room is ready. Or, Karakter is a restaurant and bar on Simpson Bay beach (right next to Mary's Boon, listed below). Karakter is an excellent restaurant and they have a service where you can have them hold your luggage for you, and they do have showers there.
I would suggest you check out the following hotels-
Mary's Boon--old style smaller Caribbean hotel, on a fabulous beach, has a pool and a restaurant onsite, as well as Karakter right next door. They do have some 'budget' rooms there, that you could probably book two nights for the price of one night at another hotel around the area. Or, if they are not busy, you could possibly work out a deal to check in early. If you decide on Mary's Boon, you could go for the day to Karakter next door, which is an excellent restaurant. Not much in the way of souvenirs you can walk to from there.
Royal Islander La Plage--this is a timeshare, so early check in would be a total no-go. Location is walking distance to a ton of different restaurants and a fair amount of shopping. This is a resort, with a lot of rooms, nice pool area, beach that comes and goes.
Atrium--again a timeshare. In the heart of Simpson Bay, walking distance to probably the most restaurants of almost any hotel on the island. Some souvenir shopping available.
La Vista Beach--smaller timeshare, family owned, not really a 'beach' there, but kind of a coral, rocky shore, but right on the ocean. There are two pools for the resort. Restaurant at the resort, along with several others within short walking distance. Shopping for souvenirs is a bit further walk, but doable.
I figure others will chip in with their recommendations.
