It is the best interests of the GOP to resist the recount, because if it turns out that they do not come out ahead after a recount then they have everything to lose. However if they are confident in the results of the recount they should not oppose it and should support it.
The democrats know that they cannot win the recount. Trump won the election by a larger margin than has ever been overcome in a recount. On the other hand, the democrats know that if they can make the recount take long enough, the states won't be able to certify the vote. It is gamesmanship.
I believe that a recount of votes should not be contested as it is in the public interest to know that votes have been counted and recorded correctly. If the votes were originally counted correctly, then in my opinion, no political party should object to this being confirmed, especially in cases where the voting outcome has been unexpected.
No, the GOP should not be contesting the recount. There is no reason. If one candidate or party requests a recount, then it should be done. If the other party does not think anything was done wrong during the election and they are confident of the outcome, then there's nothing to be worried about.