Marriage license before baby

Alright so my fiancé and I have been together for 6 years we have Ben engaged for 2 but plan on getting married next October. We have our first baby girl due at the end of December, we are getting told that it may be a good idea to just get our marriage license now before baby is here because it will make everything easier at the hospital and with insurance. We are completely open to the idea of getting it now and then having our ceremony still next October. Looking for advice and experiences in whether we should do it or not? Is it worth it to go ahead and do?

Update: when I was saying marriage license I was mainly speaking about getting married at the courthouse or whatever it may be so it’s official then have our ceremony October next year! Sorry for the mix up in my county they do it all in the same building