YehudahTzvi, while you are obviously correct and it’s even easier than that to disprove their attempts, they will usually say “well, but…” and try a different “question”.
If they CH”V succeed in planting a doubt with some other kushya they rattle off, it will not be fun to work one’s way out of that.
So it’s best to ask an LOR how to approach this, and also to know your stuff, too.
Also, as yitayningwut said earlier, I don’t know that it’s appropriate to make them leave in tears, especially if they are naive in what they are doing and sincerely believe the very mistaken notion that they are doing you a favor. I think it smacks of achzariyus, which is a particularly terrible midah.
Missionaries are people, too, of course, and I’m sure Hashem is not looking for His creations to be in tears. IF they are evil then that is a different matter, but if they’re just naive or ill-informed gentiles, I see no reason to make them feel bad, even though you have to personally reject their ideas for Jews.