Ask for a better deal
In some jurisdictions it’s possible to contact the clerk of court or the prosecutor handling your case and ask to have your offense knocked down to a non-moving violation. You also may be able to approach the prosecutor on your appearance date before your case is called.
You may have to pay court costs and a fine, but if your insurer is none the wiser it could be worth the effort.
Plea bargaining isn’t permitted everywhere.
You also may be able to ask for mitigation, pleading guilty but presenting your side of the story. Basically, you are asking the judge to lower your fine. Many states allow you to mitigate via written statements.
The outcome is up to the judge. The fine may be lowered or stay the same, and it cannot increase. Regardless of what happens to the fine, you are pleading guilty, the infraction will end up on your DMV record, and eventually your insurer will find out.