I would bet them all to balance the times I am bluffing or semibluffing. I don't have a huge checkraise range or check/call range on this board so I don't see a need to balance those ranges and not simply cbet, especially if it's an anonymous table.

The problem is that only 25-30% of our range can be happily bet this board (sets, KK-AA, JT, AQ, gutshots). Even if we add some weaker Qx, part of MP type hands and Axs with backdoor flushes (and those hands have terrible equity vs opponent's range) we can only cbet the flop 35-40%. And when we don't bet the flop it's difficult to come up with any reasonable check-defending range. My simulations shows that I end up folding 75-80% of my flop checking range. Also, it's not a solution to cbet wider on the flop, since opponent does fold only around 30-35%.

So back to my question. If we check-fold this flop so much, that our opponent should bet any two every time. Does it become more profitable to check most part of our value range and divide our range into check-calling / check-raising ranges. It's easy to create range at this board that would look something like 10% cbeting range, then after we check we check-raise 20% and check-call 20%.