One has to be very careful about how a four-fold degenerate site is defined. As you point out, there is also the need to take the evolutionary history of the sample into account. This filtering can be accomplished in HyPhy - although not in an entirely straightforward manner, but before I go through with the instructions on how to do it, may I suggest that you try testing relative rates in the context of coding models by constraining only synonymous rates? That way the notion of four-fold degeneracy is subsumed by a more general notion of synonymous substitution. You can do this type of test by choosing:

Standard Analyses
RelativeRate.bf
Codon data
Any codon model with local parameters (so that each branch has a separate dN and dS)
Constraining only the synonymous rate for the test.

If you still want to use nucleotide four-fold degenerate sites (which must be defined in terms of neighboring nucleotide positions anyway), I can tell you how to do that, but codon models still have to be involved in order to recover codon ancestral states implicit in the definition.

HTH,
Sergei