Been my experience that when one has a 'large' (heavy processing) backup that times out during backup process, it was due to one of two things ... files (like an assignment all users in course had to turn in - such as a video) or many quizzes. There is always the other ... an addon that has issues with backups ... but I think that one would raise it's ugly head when one split course backups ... and you've not reported issues with split backups.
One way to find out with split backups is to decompress downloaded backups that you've split and compare the size of questions.xml file in each un-compressed backup as well as the files.xml + files directories. That involves commmand line on local computer and you don't appear to use the command line much ... if at all. :|
Could have an issue with the quiz bank ... which you could check prior to making backups with a hidden admin tool at:
https://yoursite/admin/tool/health/
It is a heavy processing tool so you have to be patient and let it run.
After health runs, it presents you with issues that it finds and has a link for suggestions as to how to fix ... normally an sql query. That query has fixed many issues in quiz banks for me - not all - but many.
But ... do the command line backup first before diving into anything else which just might confuse things as well as make things worse with the course as it is right now.
'SoS', Ken