Third party VAT setting is not applied if adding existing product with VAT, Reverse Charge

Hmm. I am afraid No VAT is only for everything in the company setup. VAT settings is a product related item. A solution can be to create more products with in description (no vat) for instance. Do not forget to use multiple VAT codes in dictionary of course if you want to use this solution.
Another solution is to ask a modification of code for a button to a partner to apply no VAT for invoice.