Just an update for anyone following along. My plans have changed a bit, based on talking to various people and having a couple landscapers come out and tell me what they could do.
1) I decided against the initial plan of me cutting and moving all that sod back in place. Mainly because I did some reading on how much time it would take for that area and realized it was going to be much longer than I thought initially.
2) I looked into renting a bobcat / track loader with a tilling attachment to just till the sod in and push the mix around. This would be the cheapest option, and probably most fun, but I'd really rather have a professional doing the job if I can afford it.
3) One landscaper who came out indicated he could bring his excavator out, and move + level the soil underneath with the bucket after cutting up the sod and moving it to a lower area of the lawn.
4) Another landscaper has suggested bringing a bunch of new soil in. This could get more pricey, but it appeals to me most because it seems least likely to result in surprises. After getting some more precise measurements, I'll have a better idea of whether this option is affordable.