The first step would be to discuss this concern with the current psychiatrist. There should be other options as regards medications. Some mood stabilizers treat mania more easily than depression (although Lamictal generally treats depression well) but each person responds differently to each medication. If his psychiatrist doesn't have any more ideas you could seek a secnd opinion but ask them what might be the appropriate next step within their clinical discretion. Also note the use of medication and dosage are different for a younger person and you could discuss this with the psychiatrist as well. This is some information on mood disorders in general including medications that you could read for informational purposes and discuss with the psychiatrist:
when i hear that someone at this age is having BP I get depressed myself because I don't believe it. SURE pdocs dx him and we should respect their view as they are supposed to be the only expertise. But at this age the body keeps changing hormonal wise and the brain is still in its infancy. The reason why he prescribed lamictal and abilify is that both help in depression. But i would consult another pdoc. Why didn't you try psychotherapy first CBT, etc.. instead to resorting to drugs directly. Usually one gets BP at late adolescence 19 and over. Unless mood swings interfere with his life drastically i would leave him untouched by meds at this early age.
i really can't think of any other solution except of consulting another one. perhaps you could ask your pdoc also whether tranquilizers are enough for the time being, ... anxiolytic,... benzos, etc...