Ross, when making bags, just rub in thin coats of neatsfoot oil with a paper towel. Not a soaked paper towel, just one with a little on it. If the oil soaks in and disappears before 3 seconds, add another thin coat.
Too much neatsfoot will take all of the body out of a piece of leather (read floppy, soggy). It also will liberate the dye and soak dye and oil into your clothes. The famous 19th and 20th century George Lawrence leather company soaked their products in neatsfoot, and they would really ruin clothes.
Did you use a top finish?
Also, I do not dye the back of my straps dark like that, for the same reason of dye transfer. It's a lot of work to do the front-side dye work and edging without making it look sloppy, but I think it is worth protecting folks clothes such as expensive reenacting gear.
I hope this helps a little. I like the color and style of that bag.
God Bless, Marc