In Steam's settings you can add a new library folder on the larger drive. The compatdata will install to the same library folder that the games are installed on. You'd have to move the games to the new drive as well. That's the simplest way to do it
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This is precisely what I do on my current system. All games live on a separate hard drive, and Steam knows that's the default library.
My NAS is a steam library, some games will even run off of it perfectly fine. It's mostly to free up space on my SSDs with games that I'm not actively playing.
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If all of the proton stuff lives underneath a common parent directory, you could copy that directory to your larger drive, and then bind mount the new location (bind mounting effectively mounts a directory underneath another directory)