Warning ... opinionated response follows ... nothing personal!
Since you are IT, a question ... is platform Linux? Is hosting on a 'shared' plan? Do you suffer from CLAS (command line avoidance syndrome)?
Migrating the entire site is possible ... but a big factor is question 2 above.
By far, the eaiest way to 'march a moodle' (which is what you are doing) is best done via git (command line only). The first hop in the march is the time to move site under git, but that depends upon not only question 2, but 1 and 3.
Plus ... which ever way you jump, do you have enough room on server to do a minimal site backup (code + DB dump + minimal moodledata/filedir/) on the first hop? 2nd and upwards hops code + DB dump only as moodledata/filedir/ shouldn't change).
Reason for the backups ... in case the next hop fails for some reason, one can restore the last known working version.
https://docs.moodle.org/310/en/Git_for_Administrators
Git for core code only ... addons done via Moodle Admin GUI if possible.
One should check the site for functionality by going to a course that is representative of most courses in site. Thus, you and your friend will tag team ... you upgrade (and backup), your friend checks the moodle.
'SoS', Ken