I'm a Toyota fan. But. Good luck finding a good one in that price range. On the good side, a penalty box will work, as the drive is too short for the downsides to become pesky. On the downside, Toyota has made some bombs that will be in this price range (Prius in the 2010 era, anything with a 2.4L 2AZ-FE, probably something else that is down this price now).
At 5 to 10 miles, the car doesn't have to be all that great. I mean, miles per year isn't that high. I think you might be better off avoiding the Toyota tax, finding something that isn't too beat, spending half the budget (while leaving the other half for repairs and upkeep), and running that into the ground instead--while saving for something better. Like a new Toyota.
I'm shopping myself but the used market is terrible right now. I'm not sure how much I want to travel to find my next beater, and I'm becoming dubious about finding one, as the market has been constrained for a while. Supposedly prices are coming down, but only slowly, and IMO won't recover, not when new is pricing itself into ridiculous, which means demand on every lower priced used car is going to stay high. Rising interest rates means people will buy cheaper cars in order to "tote the note" and so...