The House can do three things on its own power - refuse to pass legislation until its demands are met; badger people with subpoenas under threat of being jailed at the Capitol brig, which hasn’t been done in forever; and pass a bill of impeachment.
Which is why they are doing and discussing doing these specific things. These are all the choices they have with their very limited power.
You are correct, it is blustering. The power of Congress is legislative, not administrative. In addition, the threat of arresting and holding members of the administration is also bluster.
Article I, Section 9 clearly specifies:
The Privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases or Rebellion or Invasion the public Safety may require it.