For party is full, use amp_get_new_box_id to get a box slot if the party is full (amp_get_new_party_id returns NONE if there's no free slots)
Use msh_spawn_namingmenu to spawn a nickname menu. You can see how it's used in mev_battle_catch_nickname (note how it assigns the my_monster variable to the AMP slot the monster was placed in)