Congratulations on your 3rd boy! 
Out of the 4 you have it narrowed down to, my favorite with [name_m]Graham[/name_m] & [name_m]Rhys[/name_m] is definitely [name_m]Brooks[/name_m]!
[name_f][/name_f]If you love the nn [name_m]Beck[/name_m] but are not sure about the association (as well as the similar ending to [name_m]Graham[/name_m] makes it sound less distinct) might I suggest [name_m]Beckett[/name_m]?
[name_f][/name_f][name_m]Beckett[/name_m] is adorable 
I also feel like [name_m]Barrett[/name_m] would be a good fit, nn options [name_m]Bear[/name_m], [name_m]Barry[/name_m], & [name_m]Rhett[/name_m] 