Hi
Apple Watch is not user-serviceable and Apple does not offer a screen repair or screen replacement service for Apple Watch.
If the issue is not covered under warranty or any other Apple policy (be sure to ask, with specific reference to potential battery issues), then your Apple Watch may instead be eligible for replacement via chargeable, out-of-warranty service (or via AppleCare+, if previously purchased).
Contact Apple Support (mail-in service may be available, if required), make a Genius Bar reservation or visit an Apple Authorised Service Provider to arrange for your Apple Watch to be inspected and to establish your available service options:
Out-of-warranty repair service prices vary, depending on your location and model. Prices in the US, for example, are available here:
Apple Watch Service Pricing - Apple Support
I'm afraid that no-one here can tell you what Apple will decide in your specific case. You are not talking to Apple here - this is a user-based technical support community.
In the meantime, note that Apple advises that you should not continue using a damaged Apple Watch.