Can I restore a PS3 backup to different console without erasing the existing data there?

My old fat PS3 is not working normally anymore. For some reason it does not connect well wirelessly to my controllers, does not detect wireless internet and does not detect the ethernet cable. Plus, it got the YLOD twice already so my uncle gave me his slim PS3 very cheap and it has 6 cool games in it.

I was able to make a backup of my fat PS3 using the backup utility since the USB ports seemed to work well.

How can I restore the information of my old fat PS3 onto my slim ps3 without losing either the games already saved into the slim PS3 and my copy-protected Dragon Age save files?

In other words i want to keep the games in the slim PS3 and put everything from my dying PS3 into the slim one.

I have the fat HDD, the fat PS3 itself and a backup file made from the fat PS3.

According to Sony's page on the backup utility, copy protected data can't be restored to another PS3, so it looks like you're out of luck for Dragon Age.

To get the closest to what you want, restore the backup you made onto the slim, and then do as YellowMegaMan suggests to redownload your uncle's games.

After you've transferred the data from your old PS3 to your new one, get your uncle to reenter his PSN account on your new PS3 (ie. as a different user to the one you use).

Then, logged in as your uncle, you will be able to go onto the PSN store and redownload the items your uncle bought. Once they are installed, you can then logout and login back to your own account. The games will still be playable there.