That may be true, but it's a risky thing to do just to see if someone's serious. The moment you refer me to real laywers, you have probably removed any opportunity for me to be nice to you and settle the disagreement amicably, even if you subsequently realise you've been an ass all along and want to make it all go away quietly. If nothing else, if I think you're asking me to speak to your lawyers, it's probably not me who's calling, it's my lawyers. And now everyone's paying someone's legal fees, and if you were trying to screw me, I'm pretty much going to make sure it's you who pays everything if at all possible. (Clearly this varies somewhat by where you are and how your local legal system works.)

Usually, startups are small. That means everybody knows everybody, and the CEO should know everyone. Why didn't she just forward the e-mail to their legal department/person internally, with a simple "thank you, your invoice will be reviewed"? Their response sounds like "Oh, you entered the wrong door, this is reserved for important persons. Go to the servant's door, just outside to the left".