others already mentioned it, but i would be prepared about other debris in the mound.
as far as type of grass you have and want to keep. get a shovel out get a few descent size samples and take to a nursery. ((calling before hand of course to make sure they have someone knowledgeable. i would assume all, but maybe not)) to get type of a grass seed.
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if you strip the sod that you do not want. you can get a *of course i can not remember the name of it* it is a small shovel like tool that cuts out approx 3" diameter circle out of grass ((plug)) that you can then move to area you need to grass to grow.
you can use a spade or round headed shovel to do above, but it requires extra work. vs just putting the tool down steping on it. and lifting the plug up and out and moving it to the area you want.
the area you want grass to grow. ya space the plugs a inch or 2 apart. and over the next year or 2. just let the grass spread out like it would normally.
as far as holes left in rest of your yard were you removed the bigger size plugs. refill back in with top soil or perhaps "aerate your yard" to get all the little 1/2" size plugs to use to fill in the bigger size holes. and let the grass grow back into the holes over the next year or 2.
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i don't think you will be happy with final grade, of just using the FEL bucket. it can leave some pretty bumpy lawn. and would look at a tiller, or plow,disc,harrow. or some combination of them. so you can get a nice fluffy dirt / loose dirt on top. and get everything nice and smooth.
if this was a pasture ya sure FEL and be done with it.
but if the mound is bugging you, and some smaller ruts are as well. then at least in my experience FEL was not simply enough for final grade. just left an over all un-evenness in spots.