Completing pre and checking flop are fine, but facing the big flop bet with a strong combo draw we should happily continue, not fold.
Flop Analysis
Flop K♦7♠3♠ in a limped blind-vs-blind pot gives us a very strong draw: 4♠6♠ has the flush draw plus a gutshot to the 5, so 9 spades + 4 fives minus the 5♠ double-count = 12 clean outs versus one-pair type hands. With SPR ~2.7, both betting and checking are reasonable: checking keeps our range protected in a pot with no preflop raiser and lets BB stab, while we retain the option to check-raise all in; leading a small or medium size is also attractive to start building the pot and deny equity from random overcards. Checking here is completely fine and keeps our range balanced on a dynamic board.
Flop Analysis
Now BB bets 15BB into 7BB, leaving 5BB back – functionally an overbet shove in a ~20BB effective pot. Our pot odds mean we need ~40.5% equity to continue. Against a value-heavy range of Kx and sets, our 12-out combo draw (flush + gutshot) has very healthy equity; versus a hand like KQ with no spade we’re in the ballpark of a coin flip. We’re only in real trouble when BB has a better flush draw (A♠x♠, Q♠J♠ etc.), and even then we still retain some straight outs. With this much equity at this SPR and such a strong draw, folding is too tight – this is a clear continue, and given villain’s tiny 5BB behind, shoving over the bet or just calling are effectively equivalent from an EV standpoint.