Personally, I'd install the spare and fill up the flat one. If the tire goes flat again, then it's already off and you can drop it off at a tire shop to get it repaired.
Fill it up with tater's tank. Get spray bottle filled wit soapy water and spray it around the bead and the stem. if you see air bubbles you have a leak and now you will be able to see it.. Of course if you just change it with your spare you could just take it over to the shop and have them check..
This morning I just swapped it with the spare. I am planning to go at the meet and greet tomorrow and maybe try with a compressor to see what happened.