(1) Axiom has flown three missions to ISS already with a fourth planned for later this year.

(2) Vast’s Haven-1 station module is designed to be launched on a Falcon 9, not a Starship, and will use Crew Dragon as its personnel transport. Thus, the only thing on Vast’s critical path is its own Haven-1 module. Starship is the presumptive launch vehicle for larger Vast modules intended for future stations much larger than Haven-1, some of which will be designed to rotate to produce artificial gravity.

NASA being a public government funded operation can not accept “bids” for docking.
I do think that all they can do is set a docking fee with a daily fee to stay docked and leave it to first come first served.

They might run into trouble when a private space station is constructed also.
All civilian flights will have to go to the private station. Only public open work can be done on the ISS.
The US government can not do private work with public or military personnel facilities or materials. Those jobs must be left for the civilian market.
This could also be why NASA has not yet allowed pharma companies to do their work on the ISS.
This is also why those civilians are required to do NASA work while on the ISS.

Really the only bidding should be on who fly’s them to the station and for how much. As long as Space X does not drop their prices the bids can only increase in a fight for the flights.

I could be entirely wrong but I bet something along these lines will go to court in the near future if need be.