Is it possible to deploy 2 TLS certificates on a server at the same time?

Solution 1:

Now, this certificate is about to expire in a couple months time. Is it possible to have two certificates installed to have a sort of transition period while the new keys are added to all the clients' key store?

This is a X-Y problem.

The proper solution is to add the CA certificate to the relevant keystores. Spend the months leading up to expiry adding the CA certificate, and when time comes to renew your certificate, simply replace it with a certificate issued by the same CA.