Thanks opsgeezer for that detailed info.
I flew on a full Air Europa 737-300 Y148 many years ago LBA-ACE (on a flight that should have operated ex MAN but the fire staff strike caused it to divert to LBA along with an ELAL 707!!) and it operated direct. It was a cold winter's evening however and a mid summer day might have been a different proposition!
I've been doing a bit of homework and have found that there are 3 variants of the CFM56-3 , the B1 and B2 which you mention and also the C1. Respective max thrusts are 20,000lb, 22,000lb and 23,500lb. The B1 is fitted to just 300s & 500s, the B2 to either 300s & 400s and the C1 is available to all three variants. I imagine that a C1 engined 500 would be a bit of a hotrod!
Production has now ceased on the CFM56-3, however the information that I have read doesn't make it clear whether the different variants are achieved by a mod or whether it requires an engine change (if feasible). However there are probably enough C1s in storage in the US that could be donors.
Interesting info about this winter's heavy maint requirements. I still hope they are able to look at a few other destinations nevertheless!