For 3 grand total, you won’t get anything that will be anywhere near competitive, or that won’t be in need of a ton of work and additional $.
I’d expect to spend at that much, or at least $2500 on a complete kart that would be ready to run. Then add in the trailer. A set of new tires should run less than $250, shipped to your location. Another expense can be specialty tools for karting. Your best bet would be to find a setup from someone who is getting out of the class, or the sport.
But my biggest reservation is the apparent belief that stepping into autocross in a shifter kart might be a good idea. Although easily the most speed/performance per $invested, a kart is at the same time one of the most difficult to drive and maintain. Not trying to scare you away, but if your wife is uncomfortable with the idea of autocrossing a small car like the Miata, then she will absolutely freak out in a kart. If you want to spend a couple of years in a clutch kart like a Rotax, or even a DD2, that might be a way to get in. But a 6-speed CR125 or IC engine in a kart means doing a lot of things at once, tough for even an experienced autocrosser.
Oh, and make sure that the local club has an acceptable venue that won’t break both the machine and the driver, and that karts are allowed.
Found this setup, might be something to think about?