Have them sign an NDA.

You'd be surprised how many partners are willing to collaborate so that they don't need to build a particular feature themselves. They have their own roadmaps and initiatives. If your feature really is a value add to their offering and they feel they can sell it to their customers, I suspect they'll consider negotiating.

I personally wouldn't go to the biggest players first. There may be more departmental segmentation, more execs to convince, and more red tape involved in closing the deal. They also, like you said, have the capacity to build the feature themselves. NDAs are great and legally binding, but if you don't have the capital to stand up against them in court, you may be at a loss. You really don't want to be in a position where someone else holds all of the control.

You have something to offer. You should remain in the position of control (even if it does lead to a negotiation.)

Smaller or similar sized businesses may be more willing to join forces as it serves as a win win scenario.