What the fuck is going on.
Does the company say that they can restore installed programs after the system restore? Usually it's only data, because restoring programs off of a backup onto a system restored drive is difficult, and prone to failure even if everything goes right because they put things into registry as well. It can be done pretty reliably with ghosting, but if they are telling you to back up your stuff they probably aren't ghosting(because ghosting already does that), and ghosting might carry over your problem as well.
If you keep having trouble packing it all in you might just wanna grab things like save game files and things like DF that don't need an installer. And any hard to get installers you have.
The programs themselves probably won't make it intact even if you copied them perfectly unless they are using some trick I don't know of. If you grab your save files though you'll still have those after you reinstall the program. There's a reason system restore is a last ditch attempt.
(Pretty much if you open up add remove programs in the computer's control panel and you see a game or program listed there, it's not going to make it and will need to be reinstalled)
P.S. And of couse it depends on how far back they are going on the restore. System restore often means 'back to factory' when I hear people mention it in the business and that's what this post has been assuming. If it's what windows calls a system restore you'll just lose everything that was installed after an arbitrary date, but usually data (not programs) makes it through that without needing a hard backup.