How to randomly shuffle a list that has more permutations than the PRNG's period?

There are longer-period PRNGs than MT, but they are hard to find.

To get all 3090! combinations, you need 40,905 bits of entropy. That's about 5kb. You should be able to grab a chunk of bytes that size from someplace like random.org many times with no problem. To get precisely balanced, you'll have to add some and do rejection sampling. I.e., grab 12 bits at a time (0..4095), and reject numbers higher than your current loop index. That might inflate the number of bits needed, but probably not beyond 8kb.