The British decision was taken to go early and fast with an improvised Task Force of what was in hand. The alternative (and follow up if it failed) was to go 6 months later heavy on shipping and resources and there was no way Argentine could deal with that. Chile was not needed for either and could only create an anti Chilean legacy in Argentina if it did join in but being seen helping Britain in minor ways was good for internal consumption and earning Brownie Points in arms and economic ties with Britain.
If the decision was taken to wait for six months, the international political situation would have made the attempt untenable. In OTL, even the Americans had factions that proposed supporting the Argentine action, and this was in the immediate aftermath of an unprovoked invasion of an Ally's territory. Wait six months, and world opinion will lean heavily on Britain to accept the fait accompli.
In addition, waiting six months gives the Argentine forces time to prepare for an assault, and the positions were difficult enough to crack as it was.
If six month later is following a failed attempt, then, depending on how it failed, the British forces may very well be lacking certain key elements that can’t be replaced in that time frame. For example, if either carrier is sunk, it’s not going to be replaced, and air cover goes from marginal to inadequate. If Canberra is lost before unloading, 3 Commando Brigade is stuffed. Those troops are irreplaceable, and any substitutes in Part 2 would have minimal training in landing operations. If there have been heavy casualties among the Harriers, these are not going to be replaced, and without airfields to operate from, the RAF is basically reduced to pointless Black Buck style operations, and won't be able to knock down Argentine planes.
I suspect the terrain difficulties facing an invading force are as little understood now as they were then. To be fair, 6 months later takes one from winter to summer, so the weather will not be quite so vile (and believe me, winter in the Falklands when the wind is from the south, blowing unimpeded from the Antarctic, is not funny). On the other hand, there will be damn all night time, so assaulting prepared positions will happen during daylight, which will be no fun for the attacking forces, and may well be using troops who weren't Marines or Paras, and not so well-trained in light infantry assault actions. To take just one example, how does one move supplies about on land? Good luck with using motorised transport that isn’t specifically designed for the purpose; they’re going to get totally bogged down off the few roads. Using helicopters when the Argentine forces have had 6 months to prepare air defences is likely to be painful.
I find it difficult to dismiss the difficulties of retaking the Islands, either in April-June, or Oct-Dec, in quite so cavalier a manner. Argentina may not hold the islands, but I really do not think one can give that option “no chance”. If the original poster had said that the Argentine forces probably couldn't have held the islands, I could have let that pass.
I agree that there is nothing for Chile to gain in doing anything more to help Britain than it did in OTL, and any change to this needs to be explained.