TDL 2-7: Rough 8 with a raiser and caller

[randomstumbl]

Here's a hand where I'm having trouble deciding on the best play. I think I've decided that there's one "least bad" option, but I'd like to hear anyone else's opinions and reasoning. What do you do on the third round.

I'm sure people will question the action before the first draw. My read is that the BB has a slightly better than random hand and that the UTG player has a three card draw or bad two card draw.
AAA is at seat 0 with $923.
Hero is at seat 1 with $522.50.
BBB is at seat 2 with $919.
CCC is at seat 3 with $1061.50.
DDD is at seat 4 with $193.
EEE is at seat 5 with $1224.
FFF is at seat 0.

Hero posts the small blind of $5.
BBB posts the big blind of $10.

AAA: -- -- -- -- --
Hero: 3h 8s 4c Qc 8d
BBB: -- -- -- -- --
CCC: -- -- -- -- --
DDD: -- -- -- -- --
EEE: -- -- -- -- --

First Round:

CCC calls. DDD folds. EEE folds.
AAA folds. Hero raises to $20.
BBB calls. CCC calls.

Hero takes 2 cards. BBB takes 2
cards. CCC takes 2 cards.

Hero: 3h 8s 4c 8h 7h
BBB: -- -- -- -- --
CCC: -- -- -- -- --

Second Round:

Hero bets $10. BBB calls.
CCC calls.

Hero takes 1 card. BBB takes 1 card.
CCC takes 1 card.

Hero: 3h 8s 4c 7h 6d
BBB: -- -- -- -- --
CCC: -- -- -- -- --

Third Round:

Hero bets $20. BBB raises to $40.
CCC calls…