I'm pretty sure that The Justice Department could charge him without the House's approval.

Talk to your local CIA, BATF, FBI etc. agent, and you will be informed that no Federal officer or agent can be prosecuted for any action performed "in the line of duty" without a separate and individual "enabling Act of Congress" (both houses) authorizing prosecution or a civil suit.

Only the agency for which the person acts can generally impose "disciplinary action."

The details get a bit hazy; but that's what the agents themselves are taught as part of their training.

Additionally, there are other "fuzzy words" that give a former President all the "rights and privileges of office" after "being retired" with the exception of signing the bills. Prosecution (in the US) would be very difficult without the full cooperation of Congress, and even then it's a bit muddled since impeachment has never been deemed applicable to former office holders so the rules are unknown.

Whether it should be done is one question. Whether it can be done (in or by the US) is NOT a simple matter.

John