Is there a way to keep the Portal 2 game completion background?

According to a Steam forum topic, you'll have to overwrite menu_act01 through menu_act04 with menu_act05 in Steamapps/common/portal 2/portal2/media.

Alternatively, you can go to Single Player/Developer Commentary/Chapter 10 - The Credits.

This will play the credits and the Wheatley-in-space epilogue. When you get back to the title screen, you'll have Wheatley floating in space in the title sequence. (But it won't stick around after you restart the game.)

Edit: If you start the credits as described above, you can immediately pause and exit back to the title screen, and you'll have the Wheatley-in-space menu. This is the best you're going to get on the console versions!