As mentioned above, Hertz (Spanish customer service desk, based in Dublin as far as I know) originally said to me
am afraid that it is not possible to do what you are asking for. Only the Gold Member can bee the driver of the car, if you want to add another person as a second driver, even if he is always the same person, you have to add him at the counter.

I then went back to them quoting the FAQ and Domestic Partner Definition, and, after a number of days wait, they eventually replied to me saying

would like to inform you that the option of including your civil partner as an additional driver without having to sign any contract is available in
America (other rules apply in Europe).

So it looks as this does apply to us, even though we are not US citizens or residents. The use of the term "including your civil partner", in my book, is saying that my partner as additional driver will be covered without being expressly declared on any rental agreement.

Separately, I am waiting for Alamo to come back to me on this. Their T&Cs are slightly different "Alamo Insiders Members in the United States [my stress] are allowed to have the following additional drivers without a fee:

The spouse of an Alamo Insiders Member, common law spouse, or domestic partner who has the same address on the driver's license as the Alamo Insiders Member and who meet normal renter requirements.

Immediate family member, Mother/Father, Son/Daughter, or Brother/Sister who has the same address on the driver's license as the Alamo Insiders Member and who meet normal renter requirements."

This is a bit ambiguous. "In the United States" = renting there or resident there? But that's for Alamo, when they get round to replying to me, and for the other forum.

Thanks.