The rate limits you’re mentioning apply to domains, not servers. You’re limited to 20 per week per registered domain or 5 per week for identical certificates (which is defined as certificates covering the exact same set of FQDNs), as described (in more detail) here.
Is the plan to have 5 web servers, all serving the same 1,000 hostnames, and each of these servers obtaining and using their own certificates? A better approach would be to designate one of your web servers (or a separate server for this purpose) as some sort of master server for certificates, and then push the certificates and keys to your other web servers. Using the same certificate and private key on multiple servers is perfectly fine and quite common in HA clusters.
Some concepts mentioned in the Integration Guide might also be interesting for you.