What was the lingua franca in Europe around 1700?

Actually you have answered your own question. The lingua franca on a ship would be 'Lingua Franca' - a sort of simplified French. The real French was in use all over Europe in courts and by diplomats.

The language of the European courts was French going into the mid/late-1800s. It was only in the late 19th and early 20th centuries, with the ascendancy of the British and American empires and downfall of European monarchies that English became the lingua franca for business and diplomacy. Political and business elites of the period would have considered English to be a vulgar language, with cultured people speaking French. The longstanding use of French as the primary diplomatic and business language in Europe is why we have the phrase "lingua franca".

The language of the Roman Catholic Church Mass was Latin, and had little use outside of Mass. The language used in Muslim nations, and still used today, is Arabic. Trading to the east, a trader with the East India company would need to know Chinese.

It was most certainly French which continued to be the Lingua Franca until the take over by English in the early 19th century

Although "Latin" is a reasonable answer, and probably the more appropriate if you are speaking about scholars, speaking about diplomats and commerce that's another question.

French would be my choice. The court of Spain spoke French (they were Borbons), and Spain was still pretty mightly. The French influence in Italy was enormous. It was growing as the huge population / economy power it came to be at 1750. Its influence reached as far as Russia. If you had told me "1750", French would be the obvious and unique possible choice. Fifty years before, it is still the best one.

It was and is still a difficult question. I ask myself if somebody will come up with more data.

I think Latin, if not Latin was third most used