Spanish speaking household with young kid at Rochambeau French School. Learning three languages is totally possible, kids pick up languages easily with the right environment. In my experience Rochambeau is doing a great job at immersion (both in French and in English) and academics are A+. Just keep in mind that it's a long term (and totally worth it!) commitment. It's standard in other countries to learn other languages in school.
* Support outside school even if you do not speak the immersion language, a few ideas: playdates with natives, books in the target language (borrow from libraries, buy some to create a little home library for an emotional/intellectual connection, second hand books), exposure to the culture through exhibits, shows, events (this is DC, check events through embassies, foreign cultural centers, museums...) even if the events are in English, but they show something of the culture (dance, painting, gastronomy, music, history...)... Others may offer other suggestions for language exposure outside school.
* Check school website and talk to admissions for more details and to answer your questions and concerns about immersion programs for non-native speakers.