He legally has the authorization to use the Insurrection Act. He needs no invitation from the governors. Here is a list of presidents who have used the Insurrection Act -- several times, no invitation was received for the governor of the state: (scroll down) As you will see Grant, Eisenhower and Kennedy acted without the request from the state.
https://en.wikipedia.org/wiki/Insurrection_Act_of_1807
It was changed under "W' and expanded the authority of presidential power:
Section 1076 of the Defense Authorization Act of 2006 changed the name of the key provision in the statute book from “Insurrection Act†to “Enforcement of the Laws to Restore Public Order Act.†The Insurrection Act of 1807 stated that the president could deploy troops within the United States only “to suppress, in a State, any insurrection, domestic violence, unlawful combination, or conspiracy.†The new law expands the list of pretexts to include “natural disaster, epidemic, or other serious public health emergency, terrorist attack or incident, or other condition†— and such a “condition†is not defined or limited.
https://www.fff.org/explore-freedom/article/martial-law-act-2006/
Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, That in all cases of insurrection, or obstruction to the laws, either of the United States, or of any individual state or territory, where it is lawful for the President of the United States to call forth the militia for the purpose of suppressing such insurrection, or of causing the laws to be duly executed, it shall be lawful for him to employ, for the same purposes, such part of the land or naval force of the United States, as shall be judged necessary, having first observed all the pre-requisites of the law in that respect
https://www.history.com/news/insurrection-act-thomas-jefferson-aaron-burr
Yep, all that is true. Legally, he can do what the law says.
Note I said "Penumbra": that shadow cast wide in the minds of those who will likely never hear of the law in the course of any media discussion of the legal limits of power, only the howling butthurt of those it is used against.
The problem here is that in order to do so, he would also have to round up the trolls in the media who have captivated idiot audiences for years now spewing Marxism and Trump hate, and if he missed even one, they'd be screaming to their global Marxist brothers for the UN to come in here in all its Communist glory and subdue the rogue POTUS. The gambit here is to force him to react in such a way that the force used can be used against him. These are people who are recalling one of their own because tear gas (chemical weapon) was used against them. What would they do in the media and courts if they were marched at bayonet point into detention or subjected to live fire, and how would that be spun in the global media?
Law doesn't matter.
Facts don't matter (they make up their own).
All that matters is the way the media spin sh*t, and they are, quite frankly, out of control.
However, there is hope. The media are imbalanced, and the faster you spin something with an imbalance, the sooner it disintegrates.
By forcing the Marxists, fellow travelers, and useful idiots to put the brakes on themselves before they command nothing but ash heaps, he exposes not only the true outcome of this alleged "utopia" the Marxists are pushing for, but forces their own to slow them when they don't want to stop. By forcing them to work against their own, cracks form in their structure, and those fractures in their structure, unity, and rhetoric, weaken their cohesiveness, despite their mutual dedication to "The Revolution".
In due course, the American people are wearying of these idiots and their tantrum, and if and when that patience wears through, the counter tantrum will be brutal. In the course of this, those whose stock in trade is the effluent of incontinent prevarication take on the smell they deserve. The trick is to prevent transference. If the Mayors, City Councils, Governors, and the like have permitted this, to act Federally at this point is to accept Federal responsibility for the outcome, which ultimately would become a blame fest aimed at POTUS.
Nope. Let the onus lay on the steps of those who have enabled and even perpetrated this folly, and let the people see who crafted this destruction.