In your case, install mysqltuner on both the web server and the DB server.
Why? You'll get different results due to networking.
The one that counts is the mysqltuner run from the web server me thinks.
Tweaks ... in config.php of code DB server is IP address - skip networking setting would take DNS lookups out of the loop. On the web server ... it's a client ... so there would/should be a client section in config on the DB server that might need some tweaking.
So your moodledata directory is on the web server. What's temp settings in php? Think one would want moodledata/temp/backup/ to be the one.
Consistently failing could also indicate a space issue ... partition where moodledata resides?
df -h
Do you have any .mbz files that are recent? If so, extract the questions.xml file from the backup to get an idea of question type and size.
'SoS', Ken