On p. 84 of Francis Lord's Encyclopedia of the Civil War, Vol. III, there is a photograph of 7 hatchet heads recovered from Spotsylvania, Wilderness, Cold Harbor and Bethesa Church. The dimensions of the heads are given in the text. In vol I, on p. 48 there's a less helpful paragraph description which includes the observation that "there has been practically no change in the design and size of hatchet heads in the last 100 years." (Now remember that he was writing in the early 1960s.) Among the hatchets in the photo are 3 with hammer heads, the first looking surprisingly modern. There are nail pullers present on 2 as well. Antique hatchets aren't hard to find, at least where I live, and are affordable. I bought one for $9 a few years ago. The US Army used to have an issue hatchet, I think the nomenclature was something like "1943 hatchet" that would be perfect for reenacting use. If you google this hatchet, you'll probably find pics of it. Aside from the head and handle being painted green, it's a very run of the mill tool. You should be able to remove the paint without too much trouble. A word of warning: old axe heads are just fine; old handles are not! Even the best handle is going to be dry and will crack and break quickly. You don't want that to happen while you're pounding in your tentpegs.
Pine River Boys, Co I, 7th Wisconsin
"We're... Christians, what read the Bible and foller what it says about lovin' your enemies and carin' for them what despitefully use you -- that is, after you've downed 'em good and hard."
-Si Klegg and His Pard Shorty