I know you said it's hard to come by for you, but I use lambs-wool hide, that's about 1" x 2". If its got too much height, I cut the wool down so it's only about 3/8" high and I use a rubber glove. (If the wool is left long, it sucks up a heap of dye, which is wasted in the wool).
I could be wrong, but I think the problem with daubers is the size of the wool. When the dauber is loaded with dye, 1st touch on the leather dumps a heap of dye right there, and you end up with a dark patch. All the surrounding areas are then dyed with that same dauber which has less dye on it, there's your lighter patches.
I tried to dip dye some leather once. That was mildly successful. I used a cheap oven tray that is about 3/4" deep and about 20" X 11", but the whole thing came out really dark.
Do you have space and ability to spray? Apparently it does not give the same penetration into the leather, so I have heard, but it can give great looking results.
Some people suggest a coat of pure neatsfoot oil before or after dye is applied. I think this is supposed to help the dye to even out its penetration and coverage, but I have not tried for myself yet.