You have a hair on fire emergency. You have a monster on your back. Your unsubsidized med loan of $250k loan at 6.75% according to my napkin math is accruing at a minimum $1420 per month in interest. What is going on there? Income based repayment is not your friend, but is a way for the lender to put you deeper and deeper into slavery. The only time it is worth undertaking is when you have a guaranteed debt forgiveness at the end of some period of payments, say ten years. Since you did not mention that, and are currently not paying it off, you should know what it is doing to you.
You very likely won't be making a sure 6.75% on any investments you can find, so anything else you choose will lose you money at the difference in rates. The best way to spend your extra is to try to pay down that loan. If you let that thing sit for just 3 years, at the rate you are paying it, it will be a $305,000 loan with interest of $1,720 per month instead of just $1,420. If you let it sit for 5 years, it will become a a $350,000 loan with $2,000 of interest per month. Then when your income changes and you can finally start to make real payments on it, it will be a monster. In ten years of your current payments it would be a $500,000 loan. Letting that happen is the exact opposite of the soul of this forum.
By just paying the interest every month, in five years you will be better than saving $100,000 elsewhere and will shave $100,000 off your loan PRINCIPAL. In ten you will have "saved" $250,000. That is not "pissing in the wind" and your thinking that there is any other option for investing is very stupid thinking. While I know that you may have a bright future as a surgeon, so as a result spending all of your "now" money on a debt toward the future may not be wise, since now money is tight, and paying it down is 100% of your disposable income, whereas paying it in a couple years would only take a portion of the same, you seem to be missing something if you think there is any other form of investment worth pursuing. All investing is simply paying toward the future. For you, the best way to pay it forward is to pay whatever you can toward the loan because you get a guaranteed 6.75%. That is a huge rate of investment.
By the way, I know personally what I am talking about. I went to law school and took out loans. My friends did too. Some went the deferred route. They now have $300,000 to 500,000 loans and are on the thirty year plans to pay back their loans. I will be student loan free this year, just a few months from now. It sucked, but now all my money is my own. I am free from worry. They are trapped slaving away.