The overwhelming majority of elected Republican congresspersons will never recognize the legitimacy of a Biden presidency. Their proffered reasons for not doing so will be, without exception, frivolous. But that fact will not diminish the intensity of their continuing resistance to anyone and anything a President Biden proposes.
This eventuality will play out, most significantly, on the legislative front, in the Senate most prominently, assuming that Republicans retain control there. So Senate approval of any nominees (not only cabinet appointments, but obscure subcabinet ones) requiring such approval will require that the appointments be centrist. Similarly, any legislation eventually approved in the Senate will likely be compromise legislation unsatisfactory to progressives.
Republican congresspersons will, however, not only resist legislatively, but, with a few exceptions, exhibit their continuing loyalty to The Leader and their fidelity to the lost cause, akin to the lost cause of the Confederacy, by a rhetorical strike. The strike will take the form of rarely, if ever, referring to “The President” or “President Biden.” At best, most Republican congresspersons will refer to President Biden as “Mr. Biden” or simply “Biden.” More likely they will adopt, except on the floor of House or Senate whatever insulting monikers The Leader in exile comes up with. I am voting for “The Fake President” as the usual referent. This rhetorical strike by most Republican congresspersons, part of a larger agitprop campaign, will at once satisfy their political needs and gin up their right wing base. Thereby helping keep hope alive for the restoration of the ancien regime.