Hello,
I've worked with a similar project.
The solution we use is to export a backup file to the "study moodle".
More technically, you can take a look at
https://tracker.moodle.org/browse/MDL-47776
The whole project description and documentation is at:
https://gitlab.setic.ufsc.br/moodle-provas/blocks-exam_actions/wikis/moodle_provas
It's written in Brazilian portuguese, but if you have interest, we can make an english version.
Basically, we have the same division of "study moodle" and "test moodle" that we call "remote moodle" and "exam moodle", repectively.
We have developed 3 plugins so you can link how many "study moodles" you want and automagically create "exam courses" in "exam moodle" for the courses you have at your "study moodles" (the courses are created with the same name and the enrols are synced), there are capabilities mapping for which people in each user can create these courses, monitor the exam (log in into moodle during the exam time) and take the exam.
We also use a custom Ubuntu distribution that boots up from a pendrive, a cd-rom or remote via local network, and only gives users a browser the can only access the exam moodle IPs, and user cannot access any media from the machine, unless some minor configurable exceptions.
The plugins also implements a "temporary key" feature that is generated by the exam monitor at the time of the exam that students must provide to authenticate along with its usual username and password.
It is all open source and it will be great to share it with a bigger community.
Kind regards,
Daniel