Severe regrets because I lied in my PhD application
I appreciate the question.  

I think we can all agree that you should not have lied on your graduate application.  It is to your credit that you recognize this.  That said, as a frequent thesis advisor and the recent head of Graduate Studies in my department, I feel that the main victim of this lie is...you.  If a student of mine told me they had done that, I would thank them for telling me, confirm that they should not have done it, make sure they understand that they must never falsify any academic documents again...and then get on with business.  

Cutting the funding of a successful student for this kind of indiscretion that occurred before their arrival is, for me, just not a reasonable thing to do.  It is an interesting thought experiment to wonder what percentage of the graduate students in my department had an undergraduate indiscretion as bad or worse than this.  Rough guess: somewhere between one third and one half?  It is all but impossible to know and also very subjective.   

You have to deal with your bad behavior in the past.  An unfortunate fact of the matter is that it is often not possible or not optimal to directly fix past bad behavior.  That can be an unsettling thought -- for me too, by the way.  Based on the sleepless nights you describe, I am really glad that you are talking to a therapist about this and figuring out what you can do indirectly to move on with your life.

Here's an idea: you say you are doing very well in the program.  That's great.  Is there some optional service task you could take on in the program (or the department or the school or the profession)?  I think that could be beneficial, both in reality and in your own mind.  That way, when you think about the bad thing you did two years ago, you can also think "Yeah, I feel bad about that, but I feel good about X that I am doing now."  By the way, if you do it, you certainly should put it in your CV, but if (and only if) it seems helpful for you to be a bit modest/understated about it, you could pitch it that way.

Good luck.