See, this is what I thought as well. The Higgs was well predicted and made sense in the standard model, and our measurements at the LHC seem to back up what physicists were speculating. On the other hand, BICEP2 is a much newer result and there's considerable controversy about whether it's a real result or a mistake.
So why would you automatically jump to the conclusion that the HIGGS was the problem?
The last paragraph of the Royal Astronomical Society press release seems to be agreeing with you, suggesting that an error in the BICEP2 result (or, rather, its interpretation) is the most likely explanation:
"If BICEP2 is shown to be correct, it tells us that there has to be interesting new particle physics beyond the standard model" Hogan said.
IIRC, the BICEP2 result, if interpreted as resulting from inflation, indicates a surprisingly strong inflation event. The above quote suggests that inflation with the strength suggested by other measurements (e.g. the level of inhomogeneity in the CMB?) would not create this problem.