There are several out there. I have only had a couple done yet. Blackstone Labs in Fort Wayne Indiana does a good job (about $20) and for extra you can get Dyson analysis (an expert interpertation of your results, but maybe overkill). Another lab that I have heard is good is ANA Laboratories. I would talk to a tech person at the lab, tell them what happened, and ask if the oil analysis should wait for more miles on the new oil change. See what they say. Or if you want more opinion, not to steer folks away from here, but the Bobistheoilguy site has a huge level of traffic and you will get multiple opinions on this.
What I meant about the oil light was, when you got back and poured the rest of the oil in, did the light come on and for how long. If it only came on for seconds, then it would tell me the quart was enough to maintain fairly continuous flow during the one mile trip and likely no damage was done. But I think you dumped the rest of the oil in and went home without starting it and so don't know for sure.