It's more about the speed of the kill and having some residual kill. Trying not to nerd out, but at the same time answer your question:
certainty (or sertay as generic) will kill your sedges, but it is slow. I find this is preferred, because it will actually kill the nut, which can be surprisingly deep in the soil.
MSM - the best chemical for the VBW. And it will last a few weeks after the application as a pre-emergent. VBW, every little speck of it can germinate into a new plant. MSM is the one chemical I have found that prevents this due to remaining present for a few weeks.
Sulfentrazone - will also hit the sedges pretty hard, but will wreck st augustine. Not important if you're rebooting, so it would be a good added measure.
You could mix all of them in one spray application safely.
If you're doing a kill-all, a bottle of fahrenheit will run you $40-$50 and will last you a long time. Application rate is 3 to 6 oz PER ACRE. You will want a diet scale that can read in grams to properly measure for a yard your size. Apply at the high rate. certainty, you get a 1.25oz bottle for $130 or so, which is good for an acre. They also sell these little packets cheaper, which will be a better dosing option for your yard. If you're doing a full kill, also mix in some sulfentrazone.
Give it about three weeks or so before you lay more sod. You might hit the yard once or twice during this time with glyphosate, which dissipates almost as soon as it hits dirt. You can apply glyph two days before you sod. You may want to run a tiller to break up the soil compaction and make it easier for roots to anchor. I would, but it's your call.