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We're contributing 12.5% of the pot and are only 6% to hit the flush by the river. I don't think we can value raise here because on flush value alone. I wait for the flop, if I'm 4 flushed I jam the pot for value. I'll also stick around for the turn with a pair and backdoor.

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How about this? A fair share of a seven way pot is around about 14.3%. Axs will win more than this. Izmet Fekali ran this hand in this situation 100,000 times using turbo sims. Axs won 20.4% of the pots. So not raising in this situation costs you money.
The other nice thing about this play is that players who know a little bit but not a lot will instantly classify you as a maniac. Hello action.

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I get more action then I want in the games that I play in.

I also think that raising in this spot promotes tilt. My own tilt not the tables. You're getting good odds to just limp along, might as well take them IMO.