Ohhh, no, I'm so sorry you're going through this. I deal with depression too.
First, you really need to recognize that everything you're perceiving, you're perceiving through a Fog of Depression. Your brain is attacking your self-esteem and motivation, and also causing you to worry about things that you don't need to worry about nearly as much as you think you do, and it seems even more awful because you can't see an end to your problems. There is, in fact, an end--there is a solution out there--and you can find it and your life will get better again.
This includes your perception of your past performance. You have been doing better than you think. (Yes, you have.) Don't dismiss your efforts.
Second, talk to your professor. Straightforward honesty can help you get to the root of your problems and start fixing them. If your prof is a human and not a jackal in a human suit, they shouldn't think any less of you for something you can't control. (No, you can't control the fact that you have an illness. You're not just being lazy.) There may be school policies in place protecting you from discrimination based on mental illness, which may or may not be a thing you'll see. Your worries on this count may be entirely baseless depending on what the people around you are like. Remember, you are chemically disposed towards pessimism right now and it's screwing with your head--you need to learn to recognize the thought patterns that cue you in as to when it's the depression talking.
If you don't tell your professor, then at least tell SOMEONE about your diagnosis. It doesn't even have to be someone official. Just tell someone who won't make you feel like crap about it and will maybe go to Dairy Queen with you sometimes to cheer you up. Getting out of the house/apartment/dorm/whatever is really important.
Third--It's the middle of winter; could your mental illness be worse because you've got some seasonal affective disorder on the side? If so, you should buy some full-spectrum "daylight" light bulbs. I'm talking about the kind where the color temperature is something like 5000 or 6500, instead of 2700 or whatever lower number. This helps replace sunlight in winter when your brain is missing it, and using those light bulbs in your living spaces, like in your desk lamp and so on, can help dramatically with SAD. I know it sounds like one of those bizarre homeopathic remedies, but it does work--there are studies out there and my personal experience backs them up, you can Google for them if you're interested.
Fourth, sometimes you do need to take time for yourself. Ask yourself: would the mental health professionals at your university really recommend something that would ruin your career? Where did you get the information that taking time off would burn bridges? Is it based in fact, or in anxiety? I'm not saying that your worries are for-sure baseless, but you need to double-check. You can tell the counselors about your concerns, too. And maybe you don't need to take a lot of time off--maybe a month while you try out a medication, or even less time while you just take time out to relax and take care of yourself.
I know I have a bad habit of taking things on that I can handle academically--like, I learn well enough to take them on, they're nothing I can't understand--but they're too difficult to keep up with emotionally and I have to step back one way or another. That's okay. Everyone has limits, no one's Superman, you're not a robot. But because you're not a robot, your limits WILL change over time. You've learned you're dealing with a problem; once you adjust to coping with it and find out what works, your limits will expand again. Sometimes you need time away from stress to get to that point though.
Fifth and related: take time out for self-care. Bake a cake (like this one), take care of a houseplant (green oxalis or african violets will bloom indoors), learn how to properly keep a betta fish (the people over at the forum bettafish will happily tell you how to make a betta live for years instead of months), cook yourself dinner instead of ordering takeout, read a few chapters of a kids' fantasy book each night, take a long bath with soap that smells nice, spend time with your friends, listen to music, leave the house just to buy a cookie at a coffee shop, follow a web comic (you might like this one--hold out on it, the art starting in Vol2 is pretty odd but it gets better fast).
This point might seem off-topic, but it isn't. You really do need to make sure that the thing that's been stressing you out isn't your whole life.
Good luck :) We're rooting for you!