I don't know about the TSP aspect of it, but you should be able to ask the 401(k) custodian to remove the excess $300 plus attributable earnings and return it to you. They'll calculate the attributable earnings for you, so don't worry about that. You'll pay the 10% penalty plus ordinary income taxes on the attributable earnings, but those amounts will be small so it's more of a tax filing hassle than anything.
You most emphatically don't need to leave it in. If you did, you're correct that you would pay the 6% excise tax annually until you removed it.
I haven't looked at it in a while, but I believe the 10% penalty and ordinary income taxes will be on your 2021 return. You'll get a 1099-R from your 401(k) custodian reporting the withdrawal.
If you left it in, I can't see why you'd need to amend your W-2, assuming it/they were accurate. In fact you can't "amend" a W-2. You can only work with your employer to have them issue a corrected W-2 in the rare case something is wrong on it; and an excess deferral isn't technically wrong, it's just something that's penalized in the tax code. You should just report the actual numbers on your W-2 on your tax return. If you're using a tax prep product like Turbotax, it would probably notice the excess contribution and point that out to you.