My company's plan is to have me travel to the US using my EU passport and the VWP (Visa Waiver Program) as a business visitor to perform work at a customer's office.
In the meantime I'll also get started on the process to get the TN visa for my Mexican passport. The company will be flying me to my home city for consulate appointments and once I have the TN visa I'd return to the US using the TN visa.
This sounds like a red flag to me as I don't think the VWP for business trips is interchangeable with a TN visa. In strict terms, I will be performing the same job while on the VWP as with the TN visa.
EDIT: I have travelled with the VWP in the past. The work that I'll be performing is software engineering for a customer. The length of stay is up to 1 year once I have the TN visa.