I believe when you try to use the same certificate on Edge that you have installed on your on-prem mailbox server, it will fail. The certificate on Edge must be issued from the same CA and have the same subject name as the cert on the mailbox server, but they must be different certificates.
See: https://learn.microsoft.com/en-us/exchange/certificate-requirements
"Each Edge transport server must use a certificate that shares the same issuing CA and the same subject for hybrid secure mail to function correctly."
You can use one certificate on multiple Edge servers with a SAN that includes the external DNS names used for them all.