There's no doubt your likelihood function is misbehaved here (rho near 1 is worrying) and what you're seeing are numerical problems.

My question: When it comes to rho "near 1" what about the extreme case of rho=1? I tried another biprobit estimation which shows up rho=1...with no error message. At that point, should the likelihood computations have stopped?

(just in case, here's attached the rho-equal-one output)

Artur