First of all, I'm really sorry that you're having such a hard time right now. You're going through a lot but you're making it - one day at a time. I think it's easy to allow the stress, the anxiety, the self-doubt and negative thoughts weigh heavy on you during such a crucial time.
Take a moment to focus on what you're doing right and where you are being given breaks. You had said that a few of your teachers are easy graders, which means you were given a break. So while it wasn't your best work, there isn't anything to worry about the semester is already over and they gave you a break. Yay!
The next is, despite the stress you have been under you're being proactive in seeking treatment! That shows that you're motivated and believe in yourself and your future success to get help. You are actively trying to manage your mental health issues and that's huge.
Third, you're going to have all summer to get adjusted to medication to ensure that you are on a better path towards success and getting your graduate degree.
So, while it's easy to get overwhelmed you are still taking care of yourself and obviously those around you notice and believe in you or else those teachers wouldn't have given you a break. I was undiagnosed until earlier this semester and it was rather difficult. I can relate to the overwhelming emotional drowning that you experience as you try to get yourself to swim towards productivity.
...All of this text later to answer your question...
Your professor sounds like he respects you and if he's offering you an extension and making a comment on the length... he clearly is looking out for you and your grade and you might want to take his offer. If you think you will actually make use of it.
Your question was should you take the extension? I really think that's dependent on you. Really think about if you are going to utilize the extension. If you accept it and don't use it then chances are you will only end up feeling worse and end up in a deeper depression and self-loathing state.
If you think you can add to that paper, even an additional page, DO IT. Once you start writing, it's likely you will accomplish more than you thought. The benefit to adding to this paper is not only will the teacher appreciate the additional work but more importantly YOU will feel so much better*!
*-Probably. I'm just a random person on Reddit, I know nothing. ;P
I know during this semester when I was feeling that way, I would go to a local coffee shop and listen to some music to write my papers. If I stayed at home, I got distracted MUCH easier.
Find something that works for you! Something that gets you away from wherever you find yourself to be stuck.
Anyways I would not mention it to the professor, especially if he offered the extension. I would just graciously accept if that's the route you choose.
Remember to focus on the positives, I know it's a hard thing to do right now but you're working your way towards a better tomorrow. One day at a time!
Let us know what you decide! Best of luck!!
I guess I will go study for my finals now. Haha.