Buying a seat will not relieve you from paying the pet fee.
Also, you will run into issues with four dogs. I don't know a single airline who will let one person travel with four dogs in the cabin. With your size dogs, that'd only work if you are a party of four, and then there's the fact all airlines limit the number of animals flying on any given flight. For example, DL limits the number of pets to 4 in the whole Y cabin - that means your party will take up all the capacity, and if someone else checked in with a pet before you, one in your party will have to take a different flight. This is similar on all airlines.
Korean only accepts a maximum of two dogs per carry-on bag if the animals are younger than 6 months - also, the weight limit is per carrier, not per animal (that's the case on ALL airlines).
What I suggested in your thread in the DL forum - using LH ex DFW - makes the most sense here, provided you will be four passengers, each with one dog carrier. LH only allows one animal per carry-on carrier. And they are very rigorous in their enforcement of the "needs to be able to stand and turn around" rule.