I'd have the Argies delay until May 24, and present the invasion to the people on their national holiday. This would push any invasion attempt to the next southern summer and allow both sides to organise better and would generally ramp up the level of violence on both sides.
The problem with that is that OTL tensions had begun to wrap up weeks before the initial Argentine invasion thanks to the exploits of the scrap metal merchants on South Georgia. IIRC the initial landings in South Georgia were less a part of the plan than an opportunity exploited by it, so might still go ahead. This is important because the initial British moves were in response to the South Georgia operations, not the actual invasion - in particular, two nuclear submarines were ordered south on the 29th of March. Any delay of more than a fortnight means these boats are on station and the invasion simply doesn't happen. Even if south Georgia is still delayed, an extra two months of tension may also provoke the British government to at least reinforce the garrison of the Falkands and even as little as a properly equipped battalion of paras would probably be enough to make the invasion unacceptably costly.
I think if you want to make the war bloodier the only change you need is a much easier one - take away from the Argentine leadership the delusion that a democracy led by a woman will be too weak and indecisive to respond forcefully and give them the understanding that they are going to have to fight to keep the islands - put professionals in as the garrison, not raw conscripts, make sure they have plenty of artillery, anti-aircraft weaponry and as much armour as you can get to the islands - IIRC the British got more tanks onto the Falklands OTL than the Argentinians, which is ridiculous - and most importantly appoint a general with fire in his belly to command the garrison, not a military bureaucrat.
It probably won't change the outcome, but a garrison willing and ready to fight will make life much more difficult for the British.
