The most appropriate diagnostic test to perform next is a karyotype. This patient has evidence of hypergonadotropic hypogonadism based on elevated gonadotropin levels and low testosterone level. Klinefelter syndrome is a common cause of hypergonadotropic hypogonadism and azoospermia, resulting in infertility. A 47,XXY karyotype is diagnostic of Klinefelter syndrome. Mosaic variants of this condition exist but typically present with oligoasthenospermia, testicular failure, or hypogonadism. Concomitant symptoms often include sexual dysfunction and generalized fatigue. Tall stature is a common finding. Patients with Klinefelter syndrome may fail to achieve puberty or may present after sexual maturation with azoospermia. Fertility may be achieved from ejaculated sperm, if present, or extracted testicular sperm; however, advanced reproductive techniques such as in vitro fertilization and intracytoplasmic sperm injection are necessary to achieve pregnancy. Some couples may opt to include genetic testing by preimplantation genetic diagnosis and embryo biopsy to avoid transmission of the disorder to subsequent generations. Typically, gonadotropin levels are high in patients with Klinefelter syndrome, representing testicular hypofunction. After plans for conception are completed, supplementation with exogenous androgens may be considered to prevent osteoporosis. Conception with donor sperm is an alternative fertility treatment option.

MRI of the pituitary would be needed only to rule out a pituitary mass in the setting of hypogonadotropic hypogonadism. Because this patient's gonadotropin levels are high, a pituitary mass is unlikely.

Scrotal ultrasound would identify small testicles in a patient with suspected Klinefelter syndrome, but it would not identify the cause of this patient's elevated gonadotropin levels.

Serum prolactin level would likely be normal, as Klinefelter syndrome is characterized by primary hypogonadism with normal prolactin levels. Therefore, measurement of serum prolactin is not the most useful test for this patient.