If, as claimed by the acting Sec of DHS, these are federal LEOs from CBP, ICE and the US Marshalls Service then they are all empowered by federal to protect federal buildings and facilities wherever they may be. Federal law also allows them to prevent, investigate and make arrests for any actual or reasonably suspected violation of federal laws wherever they may occur or the suspect is found.
These powers are independent of local control, and require neither invitation nor consent. Local authorities may refuse to assist in various activities but can not forbid or prevent them as long as it is in furtherance of their duties of protection or enforcement of federal laws.
The Federal Protective Service has been providing security at the entrances, within and around virtually all Federal buildings and facilities For quite a while and is created and funded by Congressional action. Im not sure that they are actually Federal LEOs with arrest powers. Also not sure what, if any, their role may be beyond building security and protection.
If the governor tried the utilize the National Guard, they would simply be ordered to federal service, taking control away from the governor. They would then be ordered back to barracks. This has historically been done a number of times when governors have tried to use the Guard to inhibit federal officers from enforcing federal law. Further, it would be impractical for local law enforcement, if they would carry out such an order, to interfere or make arrests of federal officers if the feds are acting under color of federal legal authority.
It would be up to the Federal Courts to decide if their actions are within the applicable federal law, rules and regulations. The District Court might find some of the actions outside normal practices and procedures and order changes. That would be immediately appealed.
Except for lodging complaints and filing lawsuits there is little any Member of Congress or local officials can do in the short term. The complaints need to be loud, continuous, public and well publicized. Public sentiment wont affect behavior but moving the needle of public awareness and anger is always good in the long run.
At least that would be my best understanding of the situation, based upon incomplete and unverified information.
( Hello to anyone that actually read your way all the way down here. Posts like this are long, but part of it is trying to think my way through the legal issues for myself, and writing helps me do that. Although, admittedly, sometimes to a reasoned but ultimately wrong conclusion. I leave them for whatever they might add to the discussion. )