In the interests of full disclosure, one of the main reasons I am interested in UL gear is to offset the steel I carry into the woods. . .
Although you can usually find dry fuel under the canopy of trees, there are times up here in the PNW when everything is wet, and soaked in quite a bit. When you are depending on dry fuel to cook, the ability to chop and split wood can be pretty important.
Since your main concern is harvesting dry enough wood to use a wood cookstove, as opposed to a warming campfire, I think a small fixed blade knife with a blade length of about 4", a solid tang and a flat spine would be best suited. You could probably get by with a lightweight folder for a while but batoning and chopping is pretty tough duty.
Another option would be an ultralight hatchet such as the Gransfors Bruks Mini. This is ideally suited to the task you specify but does not have the multi-use appeal of a fixed blade knife.
Moras are lightweight and inexpensive, but their stick tang can fail during this kind of use. (Many people do it all the time, but if you're counting on it, a better suited knife may be in order.)