Does my insurance require LEV testing?

Yes, in the majority of situations. Insurance companies require evidence that you have taken appropriate precautions to reduce workplace hazards. One way to demonstrate that your company conforms with health and safety regulations is through LEV testing. Your insurer may contest or lower coverage if you filed a claim for an event or occupational disease but your LEV tests were either absent or past due.
Although it is always a good idea to read the fine print of your policy, insurers often comply with HSE's regulations and want to see current maintenance records and test certifications.