3 servers, one certbot renewal?

Yes there are way to do it, I recently outlined a few things here: Backend webservers - #18 by webprofusion

To me the easiest method is to renew on a single server using DNS validation, stored the cert the use that cert (and private key) on your servers either by copying the files around or pulling them from a secrets storage service.

I do also have a certificate management service under development that's designed for these types of scenarios via https://certifytheweb.com but it's probably overkill for what you need.