Bur buttercup is a winter annual, which will ease your task considerably than if it were a perennial.

That said, all those seeds will haunt you and your lawn for years.

Some info, including how to manage
http://extension.usu.edu/rangeplants/htm/bur-buttercup

Here is a link that might be useful: bur buttercup

Your best option is to begin control of Bur Buttercup now, not next spring. Start by contacting your local office of the University of Idaho Cooperative Extension Service about having a good, reliable soil test done so you will have an idea what you need to do to begin making that soil into a good healthy soil that will be better able to grow a strong and healthy turf that will be better able to help you control this plant.
You may also want to dig in with these simple soil tests,

1. Soil test for organic matter. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. For example, a good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drainsâ too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.

so you can see what needs to be done before putting down seed or sod this fall.

Here is a link that might be useful: UI CES

I currently (Spring 2020) have a bur buttercup infestation on my property. I am using hand pulling and store bought vinegar as a control. On nice spring days I go out and spray early and its pickled by tea time. I do this before it flowers.