If your systems are exactly the same, you should be able to:
a) back up the Craft database for each system: client and 'ours'. This is your protection against errors or unknown issues, so don't skip it please.
b) if your backup isn't a kind to create a full structure and content sql file for each database, make this kind of backup also, for safety and for the actual transfer of content that you want. A tool like Navicat should make this very easy, but your database administrator will know how.
c) you're fully backed up, right?
d) on the 'client' system, delete all tables in the Craft database, or drop the database itself, in which case you'll have to re-create it,set utf8mb4 character and sort, and add back allowed users. Again, a tool like Navicat can make this easy, and your database administrator should know how.
e) with the now-empty 'client' database, execute (play back) the structure and content sql file from the 'ours' system, that you createdd in step a) or b)
f) now the 'client' system is running with the same data as the 'ours' system, so you will have successfully transferred the work they did there.
Caveat. If the client actually did some work on their system, like setting up users or adding/changing content, this would be lost.
Recovering it from the backup won't be possible, as complex record identities and relations won't be the same. The way to prepare for this is to re-key any extra work like this into the 'ours' Craft system, before you do the backups and start the above procedure. If you don't prepare this way, then you can re-key on the client system after you've moved the 'ours' data to it as above.
I think I've covered everything, but have your database administrator look at these notes and have a think about it.
Good fortune, and since you're near the beginning, probably the worst that could happen is to need more re-keying.
This procedure is the one many developers use all the time -- it's standard.
If you have doubts, please don't use this process; instead, re-key the missing work, as that will be safest.
If the procedure fails, you will have your structure-and-content sql file, and can replay it into the client database after emptying it again, to restore the current situation. Or use your other database backup restore method from step a), if there was one.
Best fortune, then.