When should an American official at any level suggest/request a foreign government investigate an American citizen?
I think the first question you have to answer is, what is the purpose of the investigation? Is it because the official believes the citizen violated the laws of the foreign country? Do we assume the country's judicial system is fair? What is the US interest in seeing the citizen investigated and possibly convicted of a crime (or suffer civil penalties)?
Another set of questions around "investigate". Is it okay for an American official to give incriminating information to a foreign government if the government is unaware of any offense? What is the US interest is seeing the crime investigated?
How about trades of information--an intelligence operative trades info on citizen A for info on foreign citizen B?
How about cases where a crime/offense perhaps has crossed jurisdictional lines, so the start of an investigation in the foreign country might start dominoes toppling and permit an investigation in the US?
Without delving further into the issues, it seems to me possible circumstances in some cases could justify a request or a passing of information. But, none of those would apply as I understand it in the case of Ukraine and the Bidens.
[update--addendum: I think the propoer course is to refer any suspicions to DOJ for an FBI investigation and possible grand jury. If there's no offense under US laws but might be under foreign law, passing information from the FBI to the foreign country is possible.]