We went thru a similar thing in FL a few years ago. You cannot sell or give away a timeshare solely in your Dad's name, as he is not alive to sign over the deed. The resort cannot take back the deed for the same reason--we thought that would be an easy fix. I do not think that the abbreviated foreclosure process allowed by FL law is applicable where owner is deceased.
The unit must either go thru probate and be titled into an heir's name (and then you can sell/give away) or you let the resort foreclose. In theory, there could be a deficiency judgement against whoever is named in the foreclosure (likely, the estate). In reality, many (most) resorts just foreclose (in rem) to get the deed back. That's what happened in our case. We were told to not have any correspondence with the resort once that process started and to not accept any certified letters--reject and mark "addressee is deceased." It took about 2-3 years. We got 5+ letters in the mail from the resort--notices of delinquency, notice of default, then notice of not being allowed to use unit. Then declined the certified ones and were sent notices in the mail by the Court of foreclosure, notice of hearing, notice of final judgement.
As Surf Club said speak to their atty about foreclosure, I'd be inclined to start there. I'd tell them there are no assets and you're declining taking Surf Club. Confirm that foreclosure is solely against your Dad's name/estate. FL timeshares do this all the time, as many elderly owners pass away. My Dad's foreclosure was actually bundled with 40 others (big resort) in a bulk foreclosure.