I also find where I have sycamore and Hazel together, as you have, the sycamore casts a dense shade and few brambles grow underneath. However, the clear ground makes a great place for squirrels to bury nuts, and Hazel can survive the dense shade. Result is under a sycamore you get quite a few new sprouting Hazel plants which can be dug up and relocated as desired.
Yes it's a conundrum, in my sycamore stand the bramble is supressed by the shade but so are most other things. I need to thin the stand by about 30% to allow the stronger ones to expand but that will bring more light in so I need to cut the brambles before they take off big time. There are just a few odd hazels growing in there which should then start to pick up.
Back to the deer protection. I found some plastic netting on ebay that seems quite cheap. Might be worth buying a roll and putting rings of it around the new coppiced stools until they grow away. It's balck so it won't look too unsightly.