You can get the FQDN of any IP address using standard tools nslookup or dig. They are also shown in the AWS Console. You can also use a free service like xip.io to make up hostnames pointing to any IP address, for example, 203.0.113.55 could be www.mysite.203.0.113.55.xip.io. No configuration needed, you just create a hostname and as long as your IP appears immediately to the left of "xip.io" this just works. I assume there are others like it.
It isn't clear why exactly you would need the FQDN of a public IP address for what you are describing... intuition suggests that perhaps you are trying to solve the wrong problem. Can you clarify this need?
Also, 5 Elastic IP addresses is only the default limit per region. All you have to do is write a brief explanation of why you need more, and as long as the description includes a reasonable use case, support will increase the limit.
https://docs.aws.amazon.com/general/latest/gr/aws_service_limits.html
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