Wow, OP, please do not put anything on credit cards if you can help it. I'm not sure if it's possible to give more terrible financial advice than that. Beware of reward cards and 0% interest offers -- CC companies are not charities and statistically they know they will make a profit. Yes, there are situations where you can beat the house and come out ahead, but these are rare.
I would set aside $8k ish over time, divided up however you like. The issue with federal loans is that you are limited in how much you can take out in any single year, so getting an extra 8-10k all at once in 4th year may not be possible. That said, if your cost of attendance (ie. your max federal loan amount) is higher than your actual expenses and you are taking out substantially less than the max each semester, then you could probably just maximize your federal loans only as a fourth year and have plenty. In general, I think most people need a pretty generous portion of their loans to live on. If you are one of these, I would take out 1200 each semester and set it aside. You have 7 loan checks before M4, so that would put you in the ballpark.
Yes, you will accrue some interest on this money, but the amount will be negligible in comparison to your overall loan debt (my back of the napkin calculation says ~$600 total in additional accrued interest if you take out a little extra each time). Federal loans also come with a LOT of repayment options, flexibility, etc. If you need a year of forbearance (no payments but interest accrues) in the future, you can do it with federal loans; not so with credit cards or private loans. As a resident, you will likely be doing IBR for your loans, and the additional accrued money from interview season would simply be part of your overall debt and thus included as part of your IBR payment. Since IBR is set a % of income, your actual payment won't change if your overall loan debt is 10k higher. If you do a credit card for your 8-10k of interview expenses, that will be ~250 /month extra just to make the minimum payments. That's 250 in addition to the ~300 /month in IBR payments.
I think 5-8k is probably good number. I would probably err on the high side just because you'll have lots of other expenses anyhow. Don't forget about board exams, moving expenses, etc. Moving for me is looking like it will cost at least $5,000 for a drivable move (security deposit, first month of rent, additional month of rent til my paycheck starts coming, uhaul rental, etc.). So if you take out too much, don't worry, there will be plenty of things to spend it on. Especially if you're a totally independent adult without Mom&Dad National Bank handy, you'll need to think ahead like you're doing and make sure you have adequate liquidity.
Check with your school also and see if there are any professional financial advisors in your area who will do pro bono work with med students. Many will do this; it would be nice to sit down with a pro and talk real numbers and personal details.