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I like a flat v 3 bet pre. you don't KNOW he's got a 2 draw, your 7 is rough, you're out of position, and you admit he might have a bit better feel than you in this game. For a lot of these same reasons (except now you know he's drawing 2) I greatly prefer checking rather than leading into him after first draw. he's never folding, you haven't improved, and you can only get raised and again, you're out of position. this seems -ev to me.
after it goes 1-2 on second draw and you don't improve I DEFINITELY like a lead/pat line here. if you get raised i would call/pat repping a middling badugi. I would check fold if he pats behind and obv lead again if he draws.
as played i'm check folding if he bets on the end.
I think that a pre flop 3b greatly increases the perceived strength of hand, and makes barreling off a snow more effective. Otherwise you are playing face up and he's going to exploit you very effectively headsup.
Also consider the number of combinations of 2 card hands vs. 3 card hands and pat hands. If someone liberally opens 2 card hands (not just A2/A3/23 but A7, 26, etc) I haven't done the math but am very confident that there are more combos of 2 card hands than 3 cards/pat hands combined, even if they open 3 card Jacks.
And I think I did a good job of describing Villain's perception of me, but I don't think his read is accurate, or at least it's dated, my hand reading skills have improved significantly in Badugi as well as the other games. I was reluctant to describe my read on him, which I can summarize in that he plays too loosely. I think he focuses on exploiting bad players by playing too many hands to use his hand reading skills to snow heavily and value bet thinly, but against strong players his hands are fairly face up and his ranges too weak.
And as played I agree, calling a bet on the river would be bad.