Hi Damon,
Have you tried checking the correlation for only those ratios where all three labels (L M H) are present?. The reason could be that finding any two pairs may have higher chances of false positives which may be distorting your real ratios and therefore resulting in low correlation.
Since you wanted H/M, why do you expect quantitation to suffer when you ignore light label?
I don't have idea about this mutilation trick.
If possible, a triple SILAC search can be done with ProteinPilot and any ratios can be calculated even post search (M/L, H/L, H/M or any other combination)
In freely available software solutions, MaxQuant can be used.