Hi Steven Williams,
As you are saying if both are using the same name you need to install the CA first and then the identity, or you can change the name of one of them and install them on the any order. These behavior only happens when using the same name on both certificates, other good options will be export the certificate as a pkcs12 so when you import this on the new ASA is only going to be one step and the full chain is going to be installed.
Hope this info helps!!
Rate if helps you!!
-JP-