It has clay/sand fill dirt and has been kind of leveled out, somewhat, with a skid steer before the barn was erected.
Based on your post and the pics you provided, it appears like your sub base is basically native soils the have been disturbed and then roughly graded flat. It would also appears that we are talking about an inch or two of cut and fill to level things up. If this is the case, you don't need to rent any expensive equipment or purchase any exotic materials. You definitely don't need a sheep's foot roller, self consolidating concrete or a backpack fogger.
If I'm correct in my assumptions, your little compactor is just fine. Use whatever equipment you have to cut down the high spots and fill in the low spots until you are level, As long as your lifts are only a few inches deep, just run your compactor over it a few times and you're good to go. If the pad is at the elevation you need, proceed with slab prep. If it needs to come up, don't get too fussy with rough grading and import a good granular material, rinse and repeat.
It geotechnical terms the loads on the soils under a slab on grade are fairly modest but it does need to provide
uniform support to the slab so be consistent with materials and methods.
Take a look at my Guide to Floor Slabs in the link below for some additional thoughts on the process. Good luck with your placement and congrats on the new shop.