My favorite would be:
Preempt: to take the place of (something).
It's in common use referring to resource competitions (like bidding).
Similarly:
Supplant: to take the place of and serve as a substitute for especially by reason of superior excellence or power.
I feel like "supplant" emphasizes more the action of taking the place after conquering (rather than the process of vying for it as in preemption.)
It would be a bit figurative but you could also say
Dethrone
or
Depose
which both refer specifically to removing people from positions of power.
The <outbid> bidder...
certainly doesn't sound right.