Step 1: The patient is perimenopausal with heavy bleeding driven by a structural lesion - a 2 cm submucosal myoma. At this age and with a clear anatomical cause, definitive surgery is favoured over hormonal trials.
Step 2: Hysterectomy resolves both the bleeding and the fibroid in one definitive procedure and is the recommended option for symptomatic fibroids in an older woman who does not desire fertility.
Step 3: Combined OCPs or progestogens only temporise and cannot eliminate the fibroid, while endometrial sampling merely evaluates the lining rather than treating the cause. The decisive therapeutic choice is therefore hysterectomy.
\[\boxed{\text{Hysterectomy}}\]