Multiple Windows Authentication Prompts after F5 Authentication

You absolutely do need the APM delegation account to have the http/ssrs.example.com SPN included. I'm assuming you already have the individual server SPNs in the delegation list (else they wouldn't work either). So, the account running SSRS, is it one single account in the AD that's running all of the SSRS services on a all of the servers? The fact that http/server1.example.com works, but http/ssrs.example.com doesn't sort of suggests that the SSRS service is actually owned by the local system and not this other account.