GRCm37-designed exon target enrichment, which reference to use?
The only possible issue with using GRCm38 as the reference is that since GRCm37 was used in the bait design there are likely a few probes that will show artificial structural variations, due to errors in GRCm37. However, these will be present in all of your mice and if you use liftOver to look at the regions around the probes then such regions should become obvious. In fact, that'd be a convenient thing to do if you want to keep your (likely better) GRCm38 alignments, flank each probe site by a few hundred bases and then use liftOver to get the GRCm37 coordinates. If you go through the result you'll be able to find the false-positive structural variants due to the baits being designed with an older reference.

Alternatively, just use GRCm37 and hope that any issues with it don't result in false-negative SV calls.