Hi Marcos. Thank for the suggestion(s).
Your first suggestion (exclude all the certificates): as gwkr57 suggests, there are a lot of certificates associated with LMI Rescue, so you're playing whack-a-mole. If you're working with a third party support desk, you'd need to tie down one of their support staff for a period of time and start repeated sessions until you were reasonably confident that you had excluded every possible certificate. They (quite understandably) don't see why they should spend the time fixing what is ultimately my problem, because I use ESET. So that doesn't seem a practical solution in my case.
Second suggestion ("Can you add *.LogMeIn.com as an allowed URL? Or *.logmeinrescue.com or whatever?") - I know how to whitelist URLs in web filtering. But we're talking about protocol filtering here. That's two distinct things in ESET. In protocol filtering, you can either whitelist by IP address, or designate a specific program a web browser (which as far as I am aware, you can only do if you know the specific path of the executable. You can't know that with LMI Rescue because it runs in a temp folder.)
Do you see what I mean? I really appreciate your help, but I don't see how to make it into a workable solution in my case.