I’d say to start, in a 2 chip/3 chip structure, i get the call, but it is still not something i’d do too often. I’d really only complete and draw 4 with a 2. When you don’t pull 2-3 good cards, you are just in a bad spot.
Anyway, after the first draw, in general, you check to the “better” draws and would check raise here since you improved so much. Since its 3,2,2 hopefully you can push out the 3 card draw and maybe one other opponent, secondly, you are sort of sending off alarm bells when you draw 4 in a 4 handed pot and lead. Then again, if this game is playing super passively (considering people and limping and overlimping, maybe it is) then maybe it’s safer to bet/call there, because having it check through would be bad. Game conditions would determine that IMO, but standard play is to check raise here.
After the second draw, going 1,2,2,2 you definitely should lead here as you did. It’s a multiway pot, but you have a great 2-7 draw and a decent badugi draw, and you have balance your ranges so you aren’t super easy to read, by not only betting made badugis and what not when you are out of position and ahead on the draws. Also, because it is a limped pot and there was no raising after the first draw, the pot is pretty damn small and people would be making a big mistake drawing two for a second time. If you check, it will check back somewhat often, and again, that is a disaster. Bet-calling here is the only move you have, so on to the third draw.
Here, you bricked, the pot isn’t very large, and you are dead on the 2-7 side. If you had a 3 card badugi like 234, 235, even 245, I’d probably call, but unless there is a great read on your opponents limping range, you can’t be calling with 3 card 7s profitably. Because villain overlimped and drew 2, he almost never started with a good 3 card, and drawing 2 and patting generally represents a good 2-7 hand with who knows on the badugi, or a mediocre two way hand like 9-9 or 10-10.
Keep in mind regarding the hero-call, or bluff options, that this pot is very small for draw game standards as everywhere I have played them, they are usually games with lots of raising pre and more action post flop, there will be better spots to call down in general.