A lingua franca isn’t decided upon, it just happens to become one because of some power its speakers hold. In the Indonesian archipelago, Malay became a lingua franca because it was used by traders. In Europe, French was a lingua franca because French held a large amount of prestige among the European nobility. Now, English is the global lingua franca because English-speaking media have dominated the global media landscape.
If you want there to be another lingua franca in Europe, that language will somehow need to attain a good reason for it to become one. You can’t just pass a law proclaiming it now being ‘the lingua franca of Europe’.
Forcing people to speak eg. German by law might work, though you’ll probably have to be prepared to coerce people into actually doing so, and thus will have to ask yourself whether that’s worth it. Otherwise, there’s a good chance people will not really give a shit about your stupid law.
You could also maybe abolish all EU level accommodation for other languages than the official language in a new federalised Europe. Then, if you want anything done at that level, you have no choice but to use the official, non-English, language. This seems like it might spur an elitist environment where only a small layer of Europeans (outside of the country from which the speakers of the official language originate) will generally be able to speak that language.
This all seems a bit fantastical, though. Unless Europeans en masse stop consuming English language media, and at the same time start consuming the media of one specific other language (thus it’s a movement away from English and toward some other language by language users themselves), there won’t be a new lingua franca in Europe.