Is there a weight limit for my mountain bike (Mongoose 29” excursion)?

Every bike has a weight limit. It's likely in the manual, but usually non-racing bikes are in the range of 250-275 officially (which is often conservative). Unofficially, people go (much) higher, but 310 pounds is probably okay (I'm not a lawyer; this is obviously at your own risk). The 27.5" Excursion has an official 250 lb weight limit according to google, and I'd guess this is about the same.

The key part is to be gentle with the bike; don't jump it off curbs, etc. and it's more likely to last. My guess is that the weak point will be the spokes breaking in the wheels, but a bike shop could handle the replacement.

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A lot of heavier riders in your weight group go for rigid/front suspension mountain bikes; think Trek 820 (the mountain bike), old Specialized Hardrock/Rockhopper's, Giant Boulder's, etc. These are a bit better at taking hits and aren't really designed with too much weight optimization in mind, so they should be good options. Used ones aren't too expensive in this class (sub 100-150 bucks, probably). Since you already have the bike, I'd probably suggest using it until (if?) something goes wrong. Then, if you haven't lost some weight, try one of those options.

If you want to google for a term, look for clydesdale / athena riders; bikeforums.net and other places have sections devoted just for them (clydesdale being heavy men, athena heavy women).