T3 (liothyronine) or t4 for bipolar depression?

Not the way treated: we don't routinely treat mood disorders, including unipolar depression, with thyroid hormone. I am not familiar with other countries doing this neither.

Thyroid & mood D/O: It is standard practice to obtain thyroid function tests for mood disorders. Hyperthyroidism can have a manic-like presentation. Hypothyroidism can cause depression. In those situations treating the thyroid condition is essential for regaining health. Even providing thyroid supplementation for low - normal thyroid function in a depressed person can often make a significant improvement. Take care

T3/T4: Thyroid hormones might work as supplement for Antidepressants, but I would not recommend for Bipolar, unless the Thyroid gland is low functioning, & even then you have to be closely monitored by an Endocrinologist.