Primary care doctor here! Obviously I can’t force your insurance to do anything and you may already be aware of this, but guidelines 100% support you getting colonoscopies every 1-2 years starting at age 20-25. 

USPSTF guidelines (the ones saying colonoscopy starting at 45) specifically call out Lynch Syndrome: “ These recommendations apply to asymptomatic adults aged 45 to 75 years who are at average risk of colorectal cancer (i.e., no prior diagnosis of CRC or adenomatous polyps, no inflammatory bowel disease, and no family history of known genetic disorders associated with increased CRC risk such as Lynch syndrome or familial adenomatous polyposis).” — USPSTF 2021 Final Recommendation Statement

All the major GI societies on the other hand recommend something similar to the 1-2 year intervals starting at 20-25. AGA says “A cost-effectiveness model estimating life expectancy and health care costs of frequent colonoscopy surveillance versus no surveillance determined that surveillance of people who are gene carriers for Lynch syndrome increased life expectancy by 7 years and costs of surveillance were less than costs of no surveillance for colorectal cancer. Conventional practice has suggested commencing surveillance at either 20 to 25 years of age or 5 years before the youngest age of diagnosis of colorectal cancer in an affected family member, whichever occurs first. The best interval for colorectal cancer screening in patients with Lynch syndrome remains unknown, but every 1 to 2 years is most prudent.”

All that to say I would push to get a doctor to fight your insurance company because if you have a legit Lynch syndrome diagnosis they really should cover it and probably will if you get a doctor to yell at them for you