I've got a lincoln 135 and used it for the whole car. Like it a lot. I've also heard some positives on Eastwood welder if looking for a lower price alternative.
Everyone has their own preferences between Miller and Lincoln. I have used both and prefer Miller. FYI-Miller owns Hobart, all parts in a Hobart are Miller. I'm not a big fan of the 110v welders, you can always turn down a big machine, but you can only turn up a small machine so far....I have a Hobart handler 190(220v) that I got at Tractor Supply for around $750. Great portable machine, and cheaper than buying a Miller. Have to agree with the Tweeco guns being superior though, my favorite welder is my Hobart(miller) betamig 2510 with a Tweeco gun. Older machine, but welds great. Also have a Miller diversion 165 tig. Did I mention I'm a fabrication junky?
Don't waste your cash on the tweeko stuff. American torch tip in Florida sells to everyone. Call them and get a gun with a spare liner "90% of feed problems are from kinked liners". Pick up a couple bags of spare contact tips to learn with. I'd run .030 wire and a trimix from your local gas supplier. You don't want to do flux core welding on a car. Flux core is designed for field welding where the gas would glow away. Even a professional welder has difficulty making a flux core weld look great. Tig welding is great but it's almost as slow as brazing, and much more expensive. It's a real art to do nice welds with one, but its a very slow process.
I'd get either a new miller 110v, or the Lincoln 110v. They are both good machines, just shop around. They sell the regulator with them. Don't forget WELDING gloves, and a power welding hood. Cyber weld has everything at the best prices I've seen. Get your consumables from American torch tip.