OP, using "stagiers" is definitely done here, it is actually quite popular way to teach teachers, and an obligatory part of the teaching degree curriculum. I find it very efficient, we have them at work and they are great. You live in Lausanne, contacting
HEP should be easy for you. Why don't you go and talk to them to see if they can incorporate you, as a regular stagier, and you pay them? I would start at the dept of English, since that's your mother tongue, rigth? The admissions might be able to help you and direct you, as well.
When you think about your license...You might be able to have some of it recognized, if you agree to take some other modules to complement your US degree,
EDK (CDIP) will decide. You might be able to teach local schools, if the level of your local language is at least C1, I think. There is a bunch of info on the requirements on their site. If you dn't speak local lingo, time to really apply yourself and enroll in some intensive class. There are almost free classes, at
UniPop Lausanne. If you enroll in HEP as a temporary stagiere, you might be able to use their langauge resources as well.
EPFL also offers good French classes. The network of universities is very helpful and flexible, I am about to take some courses at EPFL, eventhough I am enrolled at my home uni in CZ.