What Recourse Could I have When Landlord ignores addressing rodent issue when my business is governed by the Board of Health?

It depends on the language of your lease. Is the landlord responsible for this type of pest control? If so, you will need to notify the landlord of the issue and provide them a reasonable time period to alleviate it. And if the landlord fails to remedy, you may have a breach of contract claim depending on the language in the lease regarding landlord breaches. Have a business / landlord-tenant attorney review the lease and advise.

Given this is a Commercial Lease situation, the language of the Lease will control. Most Commercial Leases for Restaurants and food services require the Lessee to handle any rodent issues as the Lessee controls the property. That said if adjoining properties are also have issues, you need all tenants to address. Look at your Lease or hire an attorney to review and communicate with your Landlord. All communications should be addressed as per the Lease and if in writing, send certified/return receipt