The Taiwanese government can certainly allow Chinese soldiers to do whatever they want within the borders of Taiwan.
There are no de jure borders of Taiwan. Both the PRC (which currently governs mainland China) and the RoC (which currently governs Taiwan) (officially) claim that both mainland China and Taiwan are part of one nation and each (officially) claims to be the sole legitimate government of the whole nation. Also, the RoC used to control mainland China and was expelled from it in a way (IIUC) not completely unlike the former Ukrainian government or Obama in your hypothetical.
As far as my hypothetical goes, I think Taiwan has the right to ask the US to have US military stationed within Taiwanese borders and that the US can accept such a request from Taiwan.
I don't think that would violate Chinese self-determination in a meaningful extend.
Some of the comments on the link by James_Miller exactly six months ago provided very specific estimates of how the events might turn out:
James_Miller:
Me:
"Russians intervening militarily" could be anything from posturing to weapon shipments to a surgical strike to a Czechoslovakia-style tank-roll or Afghanistan invasion. My guess that the odds of the latter is below 5%.
A bet between James_Miller and solipsist:
I will bet you $20 U.S. (mine) vs $100 (yours) that Russian tanks will be involved in combat in the Ukraine within 60 days. So in 60 days I will pay you $20 if I lose the bet, but you pay me $100 if I win.
While it is hard to do any meaningful calibration based on a single event, there must be lessons to learn from it. Given that Russian armored columns are said to capture key Ukrainian towns today, the first part of James_Miller's prediction has come true, even if it took 3 times longer than he estimated.
Note that even the most pessimistic person in that conversation (James) was probably too optimistic. My estimate of 5% appears way too low in retrospect, and I would probably bump it to 50% for a similar event in the future.
Now, given that the first prediction came true, how would one reevaluate the odds of the two further escalations he listed? I still feel that there is no way there will be a "conventional battle" between Russia and NATO, but having just been proven wrong makes me doubt my assumptions. If anything, maybe I should give more weight to what James_Miller (or at least Dan Carlin) has to say on the issue. And if I had any skin in the game, I would probably be even more cautious.