Firstly I would 3bet preflop (as I would 3bet a wide-ish range). You could go for an induce to say from 1k to 1.2k, or just 3bet shove. The 3bet shove range is villain dependent of course. If he is opening 75% of buttons then you can 3bet shove very wide- something like 50%- profitably.
Button is making a mistake by 3x raising preflop with such low stacks- a minraise to 2.5x is probably optimal (slightly more than minraise to get bigger fe against fish). Actually I think he can only play a tight range profitably when he 3xes... (haven't worked out the maths).
Also with these stacks, calling a 3x oop is quite bad imo. Since what range of hands can you call profitably? You basically need to decide on the flop whether to go with your hand or to fold, and you only hit 1/3 of the time, and stack sizes are very awkward. So you might as well exploit his 3x raise preflop by restealing wide (and because of icm he can't even make break evenish cEV calls).
As played I would checkraise flop to 1.2k or something like that (maybe even checkraise shove since you are committed anyway- again I would checkraise shove a whole range of hands).