I didn't see that problem when I just installed DA2 (and the Origin client) onto 10.10.5 and 10.11.5.
No problems whatsoever on 10.10.5 and the game patches automatically and works fine.
There is a serious incompatibility/failure of DA2 though in 10.11.x. Notably the initial securom (false) check it performs on first startup where you have to change your system time (temporarily) back to 3/24/2014 in order to launch first. Game then works, but will never apply the auto-update from 1.0 to 1.0.4 (or whatever the patch is). They really need to fix that.
But in your case, I imagine if the game cannot write to that target location it's probably due to a permissions issue with the file.
***that's my guess.
Make sure you have a backup of your data first (Time Machine) and then access the user's Home folder (Users/Johanna/) and File > Get Info. Expand the ownership and permissions section and make sure your account is the owner w/ read & write. Then select "apply to enclosed folders". This will make some unwanted changes to the permissions in the user account as well, however the next step fixes.
Boot to Recovery (CMD+R at startup). Launch Terminal from the Utilities menu. Enter: resetpassword and press return. This launches the password reset utility. You're not resetting your password. Instead you select your OS volume (Macintosh HD) then select your user (johanna) and at the bottom there's an option to Reset ACLs. Do that and then test.
If these things don't work, I suggest removing the mentioned item: /Users/Johanna/Library/Application Support/Dragon Age II/update_gamekeys.reg (or then entire DA2 folder here inside of Application Support). The game should rebuild that anyway the next time it's launched.
***You could also test launch the game in a new login user account on your Mac. Because that Library location is user specific. That will tell us if the problem is that file (as mentioned above) where then you should remove it and allow it to rebuild.
By the way, what OS X are you running?
