Before submitting to the exclusive, carefully check your contract with the non-exclusive. If you can’t withdraw your track quickly– such as within 30-60 days, you definitely shouldn’t submit it to an exclusive. I wouldn’t actually sign the deal with the exclusive unless you have confirmed that your track was indeed removed from the non-exclusive. Otherwise you are in contract violation.
Keep in mind that many non-exclusive agreements have minimum terms of 2-3 years and some are even in perpetuity. Some libraries will still let you remove on request, even if those are the terms. You’d have to ask.
Just be careful. The last thing you want is to create a mess whereby a library thinks they have an exclusive and finds out another party is still pitching it. Even worse would be a conflict showing up at a music sup which would make EVERYONE look bad.
🙂