Just fling it as far as you can, then rake it around with a leaf rake. Your two bales will disappear fast.

I hope your soil is on the alkaline side b/c Sphagnum is pretty acidic, however you do not have enough to make any impact.

For 10,000/ft2 you need around 10 cubic yards or two dump truck loads.

Before you do such a thing, do some research into why lime is spread on lawns and how much is commonly used.

From data on the western part of your state, there is nothing said about the general make-up of the soil is such that it is overly acidic....or alkaline, and hence any measureable amount of the element LIME to make any difference is questionable.
Lime, takes a long time to move through soils. Fall is the usual time to apply it if it is supposed to do something for the soil.
You cant know that unless you have a soil test to confirm whether the lime will do any good.
You can add too much lime which may spoil results the former owner evidently gave you as a lawn.
In other words, if it aint broke.......

There is nothing to assume you have any particular type of grass and since you are in the western part of your state and undoubtedly growing a cool season grass then we might make an assumption what you have is either a one type or a combo type containing Kentucky Bluegrass and Perennial ryegrass and a type of fescue.
Such combination is commonly used in northern lawns and makes for a good tough lawn that stands up to most conditions thrown at it.

If you don't wish to get a soil test, then take a plug from the lawn....at the edge of a garden...and take it to your local full service nursery where they might identify it or at least suggest you should do what will assist its good health.

Instead of the peat moss, this fall, spread 1/2" of ..either composted cattle manure or a good topsoil over the lawn. That's all.
Then in the spring, repeat this...1/2" topsoil over the lawn early enough to allow the grass coming in to grow up and through it.

This, if repeated annually, for the next 4 years at least, will feed your lawn valuable organic matter which will help your lawn retain moisture and invite aerobic matter to also help your soil.

If you were to spread such 1/2" over the 10,000 sq. ft. you would need 15 cubic yards per layering to cover such measure.
That would require it be brought by truck to your driveway where you then would wheelbarrow it to places on your lawn where you then spread it.

Any pricing of soil is readily available by phone to your local soil dealers.

Short of that, a regular scheduling of application of lawn fertilizer (granular type) should be part of your annual upkeep.

Can you post a picture of your grass?

Its possible that you've got zoysia. It has been marketed pretty heavily at times as sort of a miracle grass and was sold in cool season areas even though it's not appropriate for them. Does it go dormant earlier in the fall than most of the neighborhood lawns and turn green later in the spring?

As far as spreading amendments goes, I agree with Jeannie's idea of spreading something like compost, but unless your lawn is lower than the sidewalks, I wouldn't spread any soil on the lawn. If you do, you'll raise the level of the lawn above the sidewalks and create problems. Adding compost or other organic matter won't have that effect because the organic content will continue to break down.