Personally i don't like the river play to costly to your stack so I would take this route so if he has hit it you save yourself about 2k in chips and still have 25 -35 bbs still
Flop I would do the same as you once he calls I would expect him to be flatting with hands like 44 to qq, AJs to AKs
So on the flop I would be following through with a bet of around 1.2k
If He shoves you still have enough chips to be in the event
If he folds you win
I think its a lot more risk free then your river shove
Especially as if I was the over guy I wouldn't expect you to have pocket
,6s,8s or
in your 4 betting range this early in a tourny, so if I was him I would personally put you on AQh Akh ak, you can't possible be 4 betting this early with 76. So if I had any of this board I would make the call from him on the turn shove
And if you had say jacks and above you wouldnt be checking on the flop with the draws out there so I think it was the wrong move there.