I like the way you asserted your preference and how you elected to depart from them in the interest of resolving the problem. That's a judgment call and as long as you think through the consequences, it's your show.
Did you give up your rights by showing your CHP? (I don't know if your state requires production in this case.) Nah. It just facilitated the identification process. You could have withheld the info....could've just given your first name....etc. But it's up to you to judge how you want the encounter to go.
And deciding to go CC at the (unreasonable, really) order of the LEO is also just a rational choice. Was your right to OC abrogated? Nah. As you say, you pick your battles.
I think you handled it just fine, although, the LEO went over the line.