I popped it in and the phone quickly recognized it and picked up signal. I tested sending and receiving messages, calling and receiving calls, all with no problems. However, the next morning I saw that the phone had disconnected and was no longer recognizing the SIM card. I did the usual restart, which worked for an hour, then I had to resort to taking the card out, cleaning the contacts, and putting it back in. It worked for another couple hours, then quit again. Whatever I try it tells me my SIM card is invalid or doesn't recognize it anywhere from a couple seconds to a couple hours after I insert it. I have tested the card in an old unlocked RAZR V3 with one of the Micro to normal SIM adapters included with the SIM cutter and it works just fine, with no connection problems even after a couple days of use.
This leads me to believe that the problem lies within the software of the Moto G itself. After a bit of research it seems like this might be a problem with Kit-Kat. I've checked the SIM contacts inside the phone with a flashlight for damage (though I'm very careful with my tech) and they all appear to be just fine. I've attached a screenshot of my lock screen with evidence of the "Invalid SIM" message highlighted in orange.
I would appreciate any help figuring this out!