Just had a look at our statistics.
Averaged over the past 23 years we have shot 39% of our muntjac in Feb/Mar/Apr, 39% between May and October, and 22% in Nov/Dec/Jan.
These numbers are influenced by the fact that there's traditionally been a large shoot on the same ground (40+ days per season), which has sometimes limited stalking access between October and the end of January.
This also means we have to keep shooting muntjac over the summer - and I see that, over the last three years, we have shot up to 44% of our annual cull of muntjac in that same May to October period.
Keep in mind also that a lot of the ground we stalk is Amenity woodland, with heavy public access, meaning that it is far easier to get out and cull deer uninterrupted in the wee hours of a Summer morning than it is during the Winter.
Basically we can't afford to take the summer off, and the landowner and farmers would likely take a dim view of it if we did. Personally with muntjac over the Summer I think it becomes more a question of picking your spots where you can intercept them. So far as antlers, our muntjac policy is to shoot on sight, so it matters little whether they are carrying headgear or not.