Why did Britain hold back during the Falklands War?

Nuclear weapons would have been majorly disproportional and would have upset the international order.

Britain couldn't apply much more force than they did without significantly weakening their European commitments.

Britain sent enough forces to do the job. That's all that a civilized nation needs to do -- they weren't trying to impress anyone else

A colonel in an infantry regiment in Germany during the cold war explained the Army's policy of 'measured response' to me. The way he put it, if one drunken Russian soldier stumbled over the border on a Saturday night, they would not start off firing nuclear cannons and sending bombers to Moscow. They would dig two drunken British squaddies out of the NAAFI bar and escort him back over the border.
Same with Argentina. As little escalation as possible. No nukes. Unless we really needed to.

According to the Just War Theory;

They didn't hold back they sent what they could. As stated they had commitments in other places. The U.S. provided logistical support to them. It takes a lot effort, material, and money to send a Task Force that size half way around the world and keep them sustained. The countries that can actually do that you can count on one hand.

why did it hold back against a hilariously weak Argentina by only sending 30,000 men and one aircraft carrier as well as a few destroyers instead of 50,000 like it did to Iraq.
If they (Argentina) was hilariously weak, why would more troops have been needed??

Actually I heard the Brits sent 2 aircraft carriers, and not just one. And the Argentine Air Force was not hilariously weak - as those on the Scheffield found out. Had the Brits lost one of their aircraft carriers, they could well have lost that war.

because it had no way to get them to the war zone

as it was, they had to use a cruise ship to get the soldiers there

The British weren't holding back, in fact Britain didn't have enough ships to carry troops so they commissioned civilian cruise ships like the Canberra and had to re commission the aircraft carrier HMS Invincible after it was sold to Australia. The “hilariously weak” Argentinian Air force and navy pilots sunk more than just a few British destroyers, not to mention ancillary vessels. The British don't have any capability to threaten anyone with nuclear weapons since they need US permission to even think about using nukes and this was a conventional war, not a nuclear attack on the UK or Western Europe by the USSR. Nuclear weapons are a cold war relic and the British keep them only for prestige but can little afford them.