Typically, other entities (your customer) can be listed as "additional insured" on General Liability policies. These types of policies are specifically created to provide protection to third parties, including end-users, who engage with your business. If you already have a General Liability policy with another insurance company, they may be able to assist you in adding this language to your COI for that policy.
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The Occupational Accident and Workers Comp policies are specifically designed to protect the 1099 and W2 workers respectively in the event of work-related incidents. If a 1099 worker were to file a claim, your customer. would not be able to collect against this policy; it's solely for the benefit of the workers.
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As an example, the Occupational Accident policy is like a harness for a tightrope walker. Its purpose is to ensure the safety of the tightrope walker while they perform their duties, not to extend coverage to external parties in the audience.