Stick only, no need for MIG down there.
Wait 'til you learn MIG. When you start MIGGIN' you will discover that you can repair lots of things that a stick-welder thinks are not repairable. I started migging on the farm 38 years ago. The MIG was a Miller, in 1980, and it was already several years old. We NEVER stick-welded anything except in the field with the stick welder on the crane truck. The stick welder in the shop just gathered dust. This was a larger farm, 2700acres, so there was LOTS of eqpt.
I still run into folks to this day who think MIG is some kind of newby welding process for beginners, that can't really make a strong weld. It's a classical notion (which I understand) that people "say" for effect, but not reality. Don't knock your MIG. Everything you buy is made with a MIG, manufacturers don't go "classical" they go for production, quality & efficiency and that's MIG.
Here's a drywall screw welded to a broken wire coat hook. Welded with a MIG (on 2 Honda generators). You can't do that with a stick until you have YEARS of experience, and only if there's no MIG nearby.
would prefer a small machine that could run off a Honda 2,000i generator in a pinch.
One EU2000i is not feasible period. I ran a MIG from
TWO Honda EU2000i generators paired, did LOTS of welding, lots of projects. I tried it on one EU2000i, just to see how low it can go.
one Honda 2000 can't run a welder. The generator shuts down quickly then pouts for awhile.
But two EU2000i can. Here's welding with two Hondas (paired) replacing a trailer jack. The Hondas are inverter generators, and when paired they arrange their phases to align properly because they are designed to be paired. With two Hondas you can weld well on the 3/16" thickness setting, but it's not really enough for the 1/4" settings (1/4" is full power for 120v). On the 1/4" settings the generators bog and then overspeed and catch up, but I don't like the start of my weld to be during a "bog". But you can, in a pinch, but you have to be mindful of your weld starts. You can "pair" three Hondas (and more) then it would be good for 1/4" on 120v - but that's 3 starter ropes to pull, and who has three Hondas?
After getting it all done, and doing a real nice job I kicked myself because the tailgate hit the new jack just like it hit the old jack.

So I cut it out and moved it rearward (and to the side).