If you can post the error messages you mentioned in the logs, that might help narrow down the problem. (If they are long, you can use a site like Pastebin, and just post a link here.)
You might need your host to help though — if something is causing the WP options table to be broken, they might be limiting resources or the time that processes can run.
If the problem really is that the database connection is open to long, you could try setting the Wordfence option “Maximum execution time for each scan stage” to 15 or 12 — this will break the scan into shorter pieces, so each one will not hold open the connection as long:
Maximum execution time for each scan stage
This may not be a permanent fix, if the host stops running processes though.