[name_m]Davey[/name_m] works for me. With your sons I’d prefer [name_m]Davis[/name_m] ‘Davey’ though.
[name_f][/name_f]Since it’s not common as a full name, I think people generally will assume it’s short for [name_m]David[/name_m]. Another thing which may or may not bother you is that it sounds like the name of a youngest child.
[name_f][/name_f]I like [name_m]Davey[/name_m] [name_m]James[/name_m] best from your list. DJ would also be a possibility if it appeals.