Idk if a narrow feed dog & plate could help eliminate some of your stitching issues.
As mentioned the diamond point or LR point needles, don't seem to pull the stitches down as deep as a S point needle.
Why can't you add the heighth on the top of the hat pattern like you want. Then just modify the top part of the pattern/cap. It will be a bit smaller at the top, but keep the headband size unchanged.
I think you are going to get puckering, regardless of the needles you use.
You can try stepping down a needle size. What needle size & point style of needle are you using?
You can massage the puckering with a hammer or bone slicker or both.
Also another thing I forgot to mention, sometimes it's the leather quality/brand that can have uneven dying/splotchiness.
I had a really nice looking imported veg-tan that did that to me in 1 part of the side I bought.
How long has the bottom example been drying? If I dip using fiebings pro dye, it takes a good 12hours, sometimes more to fully dry out.
I've had no luck using a dauber. I dip dye if I can.