To address the question:
What should the postdoc do?
Short answer: The postdoc should come up with evidence-backed rationale why they should be listened to in this situation at all. Since the cat (idea) is already out of the bag, it's too late to think about whether it's right or not to share the idea, or for the PI to take and run with it. It's a judgment call and it's up to the PI at this point. From this point, it makes sense to adopt a future-oriented perspective.
Should the postdoc meet and directly address the issue of getting credit for the idea, specifically in the form of authorship on future publications? Yes, that makes sense. But I would do so only after taking some steps to increase the likelihood of reaching a resolution in the postdoc's favor. In this regard, my answer centers on the notion of negotiating leverage.
Worst case, the PI could completely usurp the idea and leave the postdoc hanging after the contract is up. Best case, the postdoc would get an extension and co-authorship invitation for major publications emerging from the project based on her idea.
To increase the likelihood of best-case scenario, the negotiator needs leverage, i.e. other options or resources to bring to the table.
From the description it seems the postdoc does not have leverage in negotiations in this case. Such leverage could include solid leads for other positions with favorable terms for developing this line of research.
Lets imagine the PI suspects you might have solid options or offers on the table to leave the job for another position that will give you more resources to build out the program of your research (employment at will allows you to do just that, unless you care about a good recommendation from this PI). If so, they will either (a) not care and let you go your way, or (b) recognize your value and be more willing to negotiate an arrangement where they would try to make an offer that competes with the offer you have from somewhere else, thus creating conditions in your favor.
It's a gamble, but if push comes to shove, it is better to find yourself in a stronger negotiating position that truly does give you a feasible Plan B if a compromise is unlikely. Good luck!