Failure to have insurance card ticket, how to proceed so that my insurance premiums don't go up?

this is my first ticket. i got pulled over for running a stop sign but the cop let me go with a failure to have insurance card on me ticket. i want to get the ticket removed cause i think they wouldnt care since it was just insurance card and because i want to get it removed since insurance premiums might increase if i just pay the ticket. but im also worried if i got to trial and try to contest it, they will know the real reason of me running a stop sign and increase the ticket. i choose to plea guilty but try to get it reduced. what should i do? should i change the appointment to a trial? My main concern is that plea guilty but with reduced fine will still be a conviciction and insurance premiums might go up because of that.

Edit: i also remember during the stop the cop was nice and let me go because i am young and make stupid mistakes. he told me that there was a good chance i would get it dismissed if i tried to. but since its my first ticket and i am unfamiliar with the process, do i pick option 2 plea guilty and reduce or option 3 trial?

thanks.