The OP started with “if I were to work for…”(the company, not in the aviation business) etc.
He asks whether he would need to be on the books as the company pilot. I don’t see that it matters what his job description is, provided that the operation meets private operating criteria, and that he is a bona fide employee.
The key is the aircraft hire agreement ie, which entity bears responsibility for what. The aircraft owner would need to have hull and liability cover, which most responsible owners would have. However cover would be limited, and most likely only extend to damage, injury or death caused by a fault in the aircraft, or if the owner or a listed pilot is flying the aircraft.
When you hire a car, it’s usually the driver who carries insurance for anything beyond a fault in the vehicle. Not always though, because it’s common enough for a company to hire a vehicle and nominate one or more drivers. In such cases the company assumes liability for the actions of the driver(s) - if they are employees using the vehicle for company business.
In the proposed case the pilot is an employee simply using the aircraft for company business. Over simplifying?
A pilot would be crazy to take on the proposed arrangement in any role other than that of an employee, with the company invoiced for the hire.