I still find the crossbow an odd choice of weapon. A longbow is much easier to make if there's access to appropriate woods. and with some good training, more effective on every level (greater range, greater penetration, greater fire rate). The *only* advantage of a crossbow is that it can be operated by an untrained halfwit (or Frenchman, as they called them at Agincourt).
Swords, daggers, spears are effective against armour, because armour has gaps, and any such conflict occurs at close range. Whilst the tradition is much stronger in the east today, there are global traditions of martial arts designed for use against armed and armoured opponents. Classical Jiu Jitsu certainly does that, and here in Europe the texts that survive by Fiore, Ringeck, Silver - all describe such systems here. To a lesser extent, Philipino escrima does the same also - lesser because it's much more about dealing with bladed weapons than it is about dealing with armour.
Have you got a local western martial arts school, or traditional Jiu Jitsu dojo you can go and do a bit of learning from? In terms of your pax - in my experience at-least 1 in 20 of most populations have done significant martial arts training so a few JiuJitsuka of teaching standard, or an expert Escrimador isn't an unreasonable proposition. Neither needs any advanced engineering skill to manufacture weapons. Judo, if you were unaware, is just a sport derivative of Jiu Jitsu, and contains techniques perfectly valid against armour, although probably not against most weapons.
Incidentally, are you aware that the book you're writing already got published in 1874 under the title "The mysterious island". Not, in my opinion, Jules Verne's best book by a long way, but he was basically writing the same plot as you are by the sound of it.
G