So, a few things.
Is it even realistic to expect this big of a jump in my score with only 3 weeks left?
No. It is extremely unrealistic to expect anything significant to happen in 3 weeks, particularly when those 3 weeks are near the end of your studying prior to an exam. Realistically, you are looking at getting a 157-168ish on the real thing right now.
You are unusual than most of the people that post on here; you have a 3.93 GPA from a top school. That is huge. If you can get into the high 160s on the LSAT, you are a looking at T14. If you can break into the 170s, you are looking at T6 if not T3. Right now, if you get a 160, you are looking at T30/50. There is nothing wrong with T30, but the outputs are noticeably different than T6.
So now is where we come to a crossroads. You need to seriously ask yourself what your values are - what are you really trying to get out of law school. It sounds like you have what it takes to get into at least the high 160s/low 170s - but it will take more than what you are doing now. I ended up getting in the high 160s - but I put in 8 hrs/day for 6 months. I went through every single published test, most under timed conditions and at least 15 full test conditions.
Now, if you aren't even remotely interested in big law or fancy fed. clerkships, than screw it. Take the LSAT now - you will get into a great law school. But if you are interested in big law/clerkships (which have their owns set of pros and cons) or if you want to get scholarship money - it may be worth it to stick it out until February or even June.