No, i think they'd take almost on the fly. I think SM or any equivalent java application ought to be able to just offer up the new cert without kicking the application.
It's those management certs you have to worry about - those will cause an outage to fix if you don't renew them through inventory-->manage identity certs.
Certs expiring is a weird thing. Had it happen once and all the phones stayed up and were fine. That's because the cert is only presented from the server to the client to open the TLS connection - not to maintain the connection. So, once the cert expired, everything was fine as long as the phones didn't have to send another TLS CLIENT HELLO. If any of those phones bounced, they wouldn't come back.
In another case, I had a AAEP. AAEP normally generates a self signed cert. You could use SMGR signed ones if you went to the trouble of doing a cert enrollment, but if you were lazy, you could use AAEP's self signed certs. You could add the SMGR CA cert to AAEP such that when AAEP sends TLS CLIENT HELLO to SM - like when outdialing a call - that AAEP will trust the cert SM offers.
In that AAEP case, we also had to take the self-signed authority cert of AAEP and import it into SMs' trusted root CAs so that when SM sent TLS CLIENT HELLO and got AAEP's cert, our SM would be able to trust it.
Then we forgot to renew the mgmt cert on SM. That meant we couldn't manage it via SMGR anymore. The only way out of that is initTM on the SM. initTM means literally what it says - initialize all trust management - as in, flush your trust stores and key stores, and in, stop trusting that AAEP CA cert - or in your case - stop using that customer PKI cert on the SIP interface. Please generate new keys and cert requests to SMGR. After a initTM, your certs will be good again for 2 years - just as if you'd done it via the inventory - but initTM is a hammer that defaults all cert configs - which in your case might cause you an outage if you're relying on SM offering that customer PKI cert on the SIP interface.
I'd say do it all in a maintenance window anyway, but beware the mgmt certs!