OK, I (well, with a lot of help) seem to have solved it, though it may not be an elegant solution:

changing a table's title works in O16, but as far as I can see not in O07. Word on user's computer says it's 2016 but if I ask for
it returns 12 (weird or not?)

So I run
but for some reason that line resets all custom shape-names to their default, obviously causing problems referring to those in code later on (like setting a picture to a shape).
Editing the table's title at least works now.

Luckily I could reorder my subs to set CompatibilityMode to 15 after looping through all the shapes.

So, all in all, thank you guys for pointing me in this direction