I often wonder whether the compatibilist doesn't see choice as a matter of figuring something out—figuring what is to be done in the light of one reasons and/or desires. If one does see choice in this way, then it is quite unsurprising that we can assign responsibility without alternate possibilities. For the process is epistemic or relevantly like an epistemic process, and it is pretty plausible that epistemic responsibility does not require alternate possibilities.
The incompatibilist, on the other hand, is apt not to think of choice as a matter of figuring out anything. This is, I think, clearest on Thomistic accounts of choice on which choice is always between incommensurables. On these accounts, when one chooses, reason has already done all it can do—it has elucidated the reasons for all the available rationally-available options. And now, given the reasons, a choice must be made which reasons to follow. This choice isn't a matter of figuring out anything, since the figuring-out has all already been done. Reason has already informed us that option A is pleasant but cowardly and leading to ill-repute, option B is unpleasant, brave and leading to ill-repute, and option C is unpleasant, cowardly, and leading to good repute. Reason has also informed us that our duty is to avoid cowardice. And now a choice has to be made between pleasure, virtue and good reputation.