[sc_rufctr]

IMO: It's your phone and therefore your txt messages so what you do with them is up to you.

In your situation I would forward the relevant txt messages only. If you refuse they could fire you for it and HR will be on their side.

This needs asking: What were you thinking when you were engaging in a txt conversation with a customer? They are not your friends.

I use txt messages to give basic info only to my customers. Things like "I'll be there in 20 minutes" Nothing more than that and it's never personal.
If they ask me a question via txt and they often do, I wont answer it using a txt message. I instead call them back as soon as I can.
Also, If a customer makes a derogatory statement about their or my organization I make a record of it after the conversation is over.

Never let yourself get ambushed. If you can see it's a customer calling only answer it when you're ready to.
I never answer a call that is a "Private Number". I always let them go to VM. If they choose to not leave a message it's not my problem.