You are looking for a specific regulation that allows your scenario, and I'm telling you there are no regulations that prohibit it. Take your pick. There is no regulation that specifically allows it either.
Back in the good old days when you could ask John Lynch for an easy to understand interpretation, he gave this one, which just uses more words, but says the same thing you have been saying:
QUESTION: May a holder of a U.S. flight instructor certificate give flight training to a holder of a US pilot certificate outside the United States, (i.e., in France)? The French Government desires a written interpretation of the FAR's.
ANSWER: Ref. § 61.3(a) and (d) and § 61.195(d); The answer is yes, a person who holds a flight instructor certificate may give flight training to a holder of a US pilot certificate outside the United States within a foreign country, and that includes in France. That is, there would be no violation of any U.S. rules over this kind of scenario. But since this training is being conducted in France, the French aviation authorities may have their own rules governing this scenario.
The only rules that even remotely address your question are § 61.3(a) and (d) and § 61.195(d). The way these rules [i.e., § 61.3(a) and (d) and § 61.195(d)] remotely address your question is that these rules cover the authority, limitations, and qualifications of a CFI, and these rules do not prohibit a CFI from giving flight training to holders of US pilot certificates outside the United States, (i.e., in France). There are no rules in Part 61 or Part 91 that prohibit it. Only in § 61.3(a) where it addresses the pilot in command legal requirements does it state: “. . . However, when the aircraft is operated within a foreign country, a current pilot license issued by the country in which the aircraft is operated may be used.” In accordance with § 61.3(a), in order for a person to act as pilot in command of a U.S. registered aircraft that the person is required to hold a valid U.S. pilot certificate or special purpose pilot authorization. However, when an aircraft of U.S. registry is being operated within a foreign country, a current pilot license issued by the country in which the aircraft is operated may be used.
Additionally, per § 61.41(a)(2), it permits the crediting of flight training given by a person who holds a foreign flight instructor certificate provided the “. . . the flight training is given outside the United States.” However, I realize your question specifically asked whether a holder of a U.S. flight instructor certificate may give flight training to a holder of a US pilot certificate outside the United States. And again the answer is yes, provided the foreign country's rules allow it. There are no FAA rules that prohibit it.
QUESTION: May a holder of a U.S. flight instructor certificate give flight training to a holder of a US pilot certificate outside the United States in a non-US registered aircraft, (i.e., in a French registered aircraft)?
ANSWER: Ref. § 61.3(b) and § 61.195(d); Yes, a holder of a U.S. flight instructor certificate may give flight training to a holder of a US pilot certificate outside the United States in a non-US registered aircraft, (i.e., in a French registered aircraft). That is, there would be no violation of any U.S. rules over this kind of scenario. But since this training is being conducted in a French registered aircraft in France, the French aviation authorities may have their own rules governing this kind of a scenario.
The only rules that even remotely address this question is § 61.3(b) and in § 61.195(d). And in § 61.3(b) the rule applies to civil aircraft of foreign registry within the United States. You are asking for an interpretation about a CFI who is giving flight training to US pilot certificate holders outside the United States in a non-US registered aircraft. There are no rules in Part 61 or Part 91 that prohibit it.
Additionally, per § 61.45(a)(2)(ii), our rules even permit an applicant to furnish an aircraft of foreign registry to be used for the practical test. However, I realize your question specifically asked whether a holder of a U.S. flight instructor certificate may give flight training to a holder of a US pilot certificate in a foreign registered aircraft outside the United States. And again the answer is yes, provided the foreign country's rules allow it. But again, there are no FAA rules that prohibit it.
