I emailed the dev who makes Solaar less than a month ago after my most recent attempt. They said clicking and holding would be on Logitech's end, in their firmware. As the firmware stands currently, it's not possible to rebind those particular buttons and click and hold them unless you rebind them to other buttons already on the mouse. Works in Logitech Options + on Windows, does not work through Solaar on any Linux distro.

I attempted to contact the Logitech devs, but there doesn't seem to be any open avenue for doing so. I tried talking to Logitech support, but they weren't able to put me in contact or even forward a message.

So yes, I have tried this recently. And literally the person who makes the software I'm trying to use has told me that my use case is not currently possible without some sort of workaround.