Thinking about this realistically ...
You have three numbers to worry about. Either not enough deer, or the right amount of deer, or too many deer. Figuring this out isn't exactly rocket surgery, just observe the effect that the population being studied has on the surrounding environment. Note that local "hot spots" are inevitable, and may need individual attention.
For the first two numbers, management is easy: Keep a hands-off approach and simply monitor the situation, unless numbers are ridiculously low, in which case a captive breeding program & subsequent release back into the wild might be in order, after ensuring an adequate food supply for the returnees.
For the last, it's steaks & sausages all 'round until the numbers fall to whatever arbitrary number The Authorities deem "normal". After which, a routine harvest of the stock should be considered a normal part of herd management.
Sadly, however. I'm sure the PTB want exact numbers with day-to-day resolution. The only way to do this is to catch and radio-tag every single deer on the island, and do it again at the end of each and every breeding cycle to tag the newborns. Until you do that, it's just an estimate, and we can't have that now can we?
At what point does wildlife become domestic?