4bet 78s, checked/shoved the flop with flush draw and backdoor straight draw

Only info on OP was that he had 3bet in his only previously hand on the button.
I 4bet bluff to attempt to fold out the 3bet, on the flop I had a backdoor straight draw and a flush draw. Rather than chase either with Cbetting I thought if I could induce the bluff I would create a lot of fold equity on this flop by shoving into the bet. If I get called I still have a lot of outs to take down the hand. Would it usually be best to Cbet the flop or as played against an unknown?

Poker Stars, $0.10/$0.25 No Limit Hold'em Cash, 6 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.

SB: $5.16 (20.6 bb)
BB: $25 (100 bb)
UTG: $31.82 (127.3 bb)
MP: $33.25 (133 bb)
Hero (CO): $25 (100 bb)
BTN: $47.80 (191.2 bb)

Preflop: Hero is CO with 8 7
2 folds, Hero raises to $0.62, BTN raises to $1.50, 2 folds, Hero raises to $3.50, BTN calls $2

Flop: ($7.35) 2 4 9 (2 players)
Hero checks, BTN bets $4, Hero raises to $21.50 and is all-in, BTN folds

Results:

Thanks for all the previous reply's to hands I've posted, especially Nitrol for his at times direct feedback (better to hear the truth) and Metza for the time taken to give an in-depth breakdown of hands.