How to split the WordPress database?

Coming from a different angle, have you looked into adding indexes to the relevant tables to maximize efficiency? Sometimes the right index can make an amazing difference, even when the files aren’t particularly large. When they are as big as yours the difference could be all you need.

You mentioned that you host the server yourself, so you have access to the MySQL configuration. MySQL tuning can do amazing things, this is another area to look into.

Also, you could move your MySQL storage to SSD which will improve performance noticeably. You didn’t say whether that had been done, and it’s not very expensive.