I am definitely checking the flop. We are not going to be a favourite when we GII on the flop and we could be in bad shape. I also prefer to overlimp pre.

The thread title seems misleading as you don't have an overpair on this flop.
Probably quite close preflop between iso and overlimp.
Bet anything else than pot is not good imo. You want to maximize your fold equity if you decide to go for the bet line: your equity is ok but your showdown value is very low.
The benefit of checking in this kind of spot, is to keep dominated draws in the pot. The problem here is that you also block 2nd nut flush draw, and the T high flush draw... which reduces a lot of dominated draws in your opponents' ranges.
I slightly lean towards potting it (and calling ofc) with this specific combo - and probably check when your K is not Kd. It probably does not make a big difference, mostly detailing this to show you the kind of way you should be thinking about blockers and how they should affect your strategy.
Check/folding is not part of the plan.

what is the SPR on the flop? if it is 1.6 as op hints then betting might be the best play

1.5, why do you mean that?

if it was SPR=2, i.e, what will be the factors that counts towards check more than bet?

yes the higher the SPR the less inclined we should be to stack off with this hand

in general the lower the SPR the more inclined u should be to stack off

Preflop is more tempting to isoraise with your actual hand.
Postflop, you block KKxx, there is never really any AAxx against you, so with your overpair and some decent backdoors I'm probably betting.
You may want to cbet small, and fold in case more than one player is getting it in. I would cbet call if only one of them shoves. I'm definitely not ecstatic about it but don't really think there is anything else we can do.