the answer is VERY. because if Japan does not led its support to the US, the US may as well not engage China over Taiwan. without Japan, the US will have no air and naval base within meaningful proximity to Taiwan.
From what I can see, both the US and Japanese governments have acknowledged that there is no point in the US going to war with China, unless the US has access to bases located in Japan.
So the logical counter is to persuade Japan that it shouldn't join the US in any war.
That is feasible because all of Japan is within 1300km of China. That is within reach of heavyweight fighters like the J-20 and J-16, shopping with Tomahawk/JAASM class
Cruise missiles.