I would definitely leave the zipper in as the neckline doesn’t accommodate for head room minus the zipper. I am wondering if you have some fleece similar to that of the chosen fabric for you to experiment with the zippers just to see how it will go as opposed to how you envision it? It might help you to make the design choices that are best suited to you sewing abilities and the needs of the recipient.
The windcheaters and hoodies tend to have the zippers with rather large teeth. I would suggest experimenting with a similar weight fleece with the invisible, lapped and chunky zipper to see what looks best.
As an alternative, if you didn’t want to do the zipper down the back, you could attach one under one of the arms at a side seam. Just as suggestion.