I think Spain was angling to get into NATO at the time and was very pro Britain. The task force would not be able to operate in the Falklands winter by the admission of its own commanders. They worked backwards from an end date of June 15, after which the troops, planes and ships couldn't fight, this is where they came to the April 21-25 landing window.
Had push come to shove, they could have done it, although the greater attrition on ships and limited flying conditions would have been very noticeable; it would probably have required a number of ships coming straight off other operations to delay their departure until fit for the rigors ahead, more ships involved (as replacements); doubtless also changes in strategy to limit the amount of time the Task Force was at sea in the area. Don't forget, the RN was well used to working in similarly 'choppy' seas in the northern hemisphere in winter months. It's very purpose was dominated by the need to provide forces for the North Atlantic region throughout the year.
OTOH, unless the Argentinians had based jet fighters at Stanley, their aircraft (and to a lesser extent their ships) would have been similarly handicapped by poor weather.
As happened OTL, the weather was often very poor - so much so that large frigates couldn't make headway so the carriers had to continue to their positions independently.
By mid-June OTL many ships in the Task Force were falling to bits and in dire need of refit - some of the ships were developing hull cracks, in addition to guns and systems being worn out - and had the objective not been achieved within a month or so of that date it would have seriously compromised their ability.
(The Navy would probably also have presented the case as "we need this done by such and such a date" for various reasons, partly to stop the politicians dawdling once they had been despatched because you can't keep a huge force like that at sea for too long.)