Ryno,
Did one at our field about 10 years ago, exactly as you described: rented a sod cutter and made about 6 passes. All by myself. Anyone who has used a sodcutter knows what it does to you. My arms and shoulders didn't stop vibrating for about a week!
It can be done. Just don't cut the sod too deep so the pieces will be lighter to move and the track won't be too deep (you'll need less product to fill it in). Use the cut sod to fill in other spots around the field, if any.
Then, also as suggested, put down landscape fabric. Even a couple of layers if you can. Even then, weeds will find their way through. But it will be much better if you put it down.
Then use limestone or gravel. Not as nice looking as most warning track material, but much less expensive. Good luck!