I'll throw out a couple of possibilities:
This is probably a silly question, but do you have any mods enabled? Those can mess up your kit, even in single player...
Have you tried only selecting a single recon team member, instead of a group?
Does it happen in any of the other missions? Can you try selecting the "recon" game type from the custom mission menu - and does the same problem happen again?
If all else fails you could cheat by typing the chat key and then theshadowknows, which makes you invisible, but I know that's a pretty unsatisfactory solution.
My game used to crash periodically on that level before the patch came out, so there was obviously something screwy with it, but I haven't heard of this particular bug before.
Sorry I can't be of more help. I'm sure someone else will chip in, they're very knowledgable, except when they're busy abusing one another
Toby
