For Mediæval European history I'd go with Latin. Latin was the written language for most of the period - Abélard and Héloise even wrote their private correspondence to each other in Latin. Then there is the problem that what was written in the vernacular is barely recognisable as the modern languages. So even if you would speak French or German, the sources will be in Middle French or German - compare Geoffrey Chaucer for a taste of the effect. On top of all that, there is the problem that neither modern language was standardised. All that great French literature is in Occitan, which is not the basis of modern French.
For secondary sources however, French is excellent. France was a demographic heavyweight in the Middle Ages, central to the European Middle Ages. Consequently, there is a wealth of mediæval research published in French.
Either modern language does have the advantage of being a living language. You can watch some movies, read some books, travel around, study abroad for six months, get a foreign girlfriend. It's all slightly more fun than the mostly esoteric ability to read Latin.
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