I don’t blame you not wanting to use those wood screws for this!
Gluing and screwing would be the best method, but gluing and nailing would be adequate. I personally always end up nailing and never had an issue. Much faster that way. But I do use a framing nailer…..not a trim gun.
If you do use screws I would still use deck screws. they have them in 1 5/8″. A bit longer than you need them….but shouldn’t hurt anything