Is this Cat. C alternate airport also a destination in your network?
If it is, and the crew for the flight are properly briefed pre-flight to hold this airfield as an alternate, meet the OM requirements to operate into this specific Cat. C airfield as a destination airport, then why would this same Cat. C airfield not be suitable as an alternate? Is it somehow worse to file as an alternate rather than to fly there as a destination? (and I make an assumption it is a usual destination in your network)
Well, an argument could be made not to use it as an alternate due to additional pilot workload during a diversion and the challenges this airport might have. That might be countered by alternate sector length, i.e. if they have sufficient time to during the diversion to establish in the cruise and brief as they would during a normal arrival to this airfield, then maybe this isn't an issue.
If the Cat. C alternate airfield is not one of your usual destinations, then there is an argument that supports not using it due to operating familiarity, if you have a choice- which you tell us you don't. In which case you carry the burden of having to ensure your crews meet the training requirements and that they are checked out for this Cat. C alternate airfield- and all the crewing and cost issues that goes with that.
One assumes that your CP or CTC has reviewed this operation and considered the Cat. C alternate issue and approved it as such.