It as, as you say, an audacious suggestion, but I fear on this, you may be reading too much into things.
The biggest issues I see with this theory are motive, ability and timing.
Motive
What would be Beijing's motive in effectively pushing Argentina and Britain into war? As of yet unproven oil and gas reserves that no one has even tried to exploit, and which are located about as far from China as it could get? Just doesn't fit.
Ability.
Argentina has been woefully under-investing in its military. It is simply not possible for them to launch a successful opposed invasion of the Falklands, and keep them against the British without a comprehensive overhaul of its entire military.
The resources and time needed to do that would be enormous, and such a massive undertaking would surely not go unnoticed.
Not only would the UK have ample warning and time to build up their own forces in counter, the US would surely see that as a gross threat.
I mean, the chickenhawks in Washington are getting all worked up about loosing the ability to being able to beat China up on China's doorstep whenever they feel like, you think China effectively building, what would be a Chinese strategic ally, into a military powerhouse in America's own backyard would go down well?
I see the most likely American reaction to China arming up Argentina in any meaningful way would be a second Cuban Missile Crisis like event.
Here is where the ability factor comes in.
If America gets so concerned as to directly involve itself(the British would undoubtably do all they can to make American feel as threatened as possible by China arming Argentina to the teeth), not even with the Falklands, but with containing the 'Red Argentinian Threat', what could Beijing do about it?
Even the USSR had to ultimately back down in the face of American nuclear brinksmanship, and unlike the USSR, China does not have nuclear parity with American, or anything like that.
Nor as yet does China have the power projection capabilities of the USSR in relation to America during 62.
China is never going to actively and willingly put itself into a stand-off it knows it could not win.
This is where timing comes into play.
The only way any of those would make sense if the balance of power is such that China has managed to successfully push American influence out of Asia completely, secured its sea lines of communication with the Middle East, African and Europe from all possible interference, and is actively looking for a pretext to establish a foothold in South America as part of an overall containment strategy against America.
Even if China has that as its long term goal (and there is zero indication and little reason to suggest that it is), we are looking at a minimal of decades, maybe even centuries before the global geopolitical situation shifts so much short of some completely unforeseeable and total calamity befalling the US.
Overall, this theory seems far more like the over-active imagination of a western poker playing mind thinking of big betting worst case scenarios, rather than the patient, focused yet all encampusing long-term thinking typical of weiqi playing Chinese strategists.
The focus is also on how to screw British instead of how to benefit China.
While playing spoiler for the sake of screwing an opponent over can be a goal in and of itself, to be frank, Britain just isn't big enough of a power or problem for China to go to such lengths and take such risks just to screw over.
The most likely scenario is actually one you already seemed to have dismissed - the AIIB.
However, I think you may have the roles and order slightly mixed, which is probably why it didn't make a great deal of sense and why you first rejected it.
Rather than China using possible arms sales to Argentina as a threat to coerce the British into joining, it seems far more likely to me that the British listed not providing Argentina with certain arms as a key condition to the UK breaking ranks with America and joining the AIIB.
Since the AIIB is of far greater import to Beijing's plans than a few measly billions in arms sales, it is a deal Beijing would have been happy to take.