Dunno, in this situation Japan holds all the cards and the USA and potentially the Russians and the Chinese are both in deep shit.
Okay, let's follow that logic, shall we? Japan actually doesn't hold as many cards as one would think. The whole Gate fiasco and the how they've managed it by not allowing even a token international force to see the other side has made Japan very unpopular in the eyes of the international community. Aside from CIA douche baggery the US really hasn't done much to put additional pressure to bear. But if Japan tries extortion, then the gloves come off.
This is not helped by the fact that it seems relations between Japan and the US are rocky in the show, when infact they've been fairly good these last few years in RL. Now that's an angle that can be played in the story as it presents opportunities.
Now, according to the JSDF, Japan could survive if they seize the natural resources of Falmart but this poses several problems.
1.) The gate is a huge bottleneck and the constant flow of shipments too and from Falmart would likely be a strain.
2.) Hardy is a fickle Godess. She keeps the gate open because she enjoys the thrashing the JSDF is giving the Empire. What happens when she gets bored of her new toy and closes the gate? JSDF is going to find itself without its new resource cash cow, and now an international pariah. And without the Stargate nor the SGCs fleet, it has no way to sustain its position or get home without asking for US help.
3.) International condemnation. This is especially bad in Asia, many of them who remember the last time Japan started seizing territory and resources for its own good. Especially as Japan is rearming again. There would be a very real concern of Japanese Imperialism that would see most of South East Asia, and especially China become hostile overnight. I'm pretty sure even the Russian Federation would be wary of this too.
4.) Alien threats. If Falmart is in the SG Verse in this story that means it's under threat from the Goa'uld or Ori. And even post Ori there is still the threat of the Lucian Alliance or hostile Jaffa elements. How do you think the JSDF are going to handle an enemy who controls the orbital high ground and csn match if not surpass them in technology and military might? And with no backup from the US to contest orbital Supremacy, it ain't looking good for the Japanese.
Of course all of this is dependent on things going fairly smoothly for Japan on Earth and in Falmart. Remember when it was mentioned that Japan would survive cut off from the international community? Well survive doesnt mean thriving. How long before the Japanese economy tanks? How long before people start losing export markets and lose life savings and national wealth? How long before they're denied food imports? How many hardships will they endure before voting in a new government that will change course? It's not an unreasonable position to consider.
This also of course depends on things in Falmart going smoothly. Zorzal was an idiot and so blatantly obvious you could see what he was doing a mile away. He was also no real threat militarily either. What if the Lucian Alliance decided to support him by giving him weapons and military support? Now suddenly the JSDF find themselves facing a credible threat they can't easily trounce. And now they need help....
...From the very people they tried to extort and blackmail.
And these are just off the top of my head.