If I were you I would talk to them. I don't know what country you are in but in my country they are very accommodating. When I spoke to the trainer the other day she said I *needed* to pass and she will do everything she can to support me to get over that line. So go talk about the depression, and see if they can give you some extensions, refer you to other supports, or anything like that.
I have exactly the same issues with the dissociation interfering with my ability to work.... I might have the intention of working on something all day and then find myself doing other things at multiple points throughout the day... bring myself back only to find myself gone again.
The other thing to work on (don't know if you see a therapist or not) is developing system communication or trying in some other manner to get alters to not switch out. Unfortunately for me one of my triggers is failure, so when the study gets hard - boom. Out comes a certain one of us to wallow in depression and not do work. Grrr.
All I can say is try and attack the issue from different angles and with the support people available to you to see if you can come up with some solutions. And you only have to give people as much info as you wish to (going with the depression is okay).