I thought about your problem this morning. I think your approach is reasonable, although I am not an expert with rate models.solved it...
I work on both the curves for model calibration:
part1: squared error in swaption prices.
part2: matching P(t, t+1) from the analytical and the model formula
Of course, I lose some information on the swaption vol part, but this seems to be a reasonable approach to deal with the two curves.
I think that your approach hypothesizes that the EIOPA process is perfectly correlated to the underlying rate of your provider data. Having a look to how are built this two curves might help you validating this hypothesis, and ease communication toward your client / boss / advisor.