You've already gotten some good answers, I just wanted to point out:

I have to do something on a live site, so I cannot test it first, so wondering if you can tell me if this would work

The most common solution to this nowadays is to set up a test environment that mirrors the production environment. Of course it's a bit of work, but one has to weigh this against the risk of bringing down the production system because one is using it to test things. In any case, you could answer your question yourself by setting up a SSCCE independently of a test system. However:

I saw somewhere online, someone used code to check if subroutine existed, before calling it.

I think this is a bit of a strange requirement, could you explain a bit more? Why not just do require "/path/to/the/file/SendMultiPartMemberEmailMimeLiteSendGrid.pl"; and be done with it? Or do you have multiple versions of the sub coming from different include files, or something like that?