He must have been eating a lot of pizza lately, mustn't he?
This sentence along with its question tag and question mark is perfectly acceptable, and grammatical. The Present Perfect Continuous tense is also used for actions that happened in the past and are repeated in the present, the action needn't be continuous, it may be interrupted or restart again.
For example, I have been marking essays since 9 o'clock this morning. If the statement is uttered at 5 o'clock in the afternoon, it doesn't necessarily mean the speaker didn't take any breaks between 9 a.m. and 5 p.m.
She has been taking the train to work since she sold her car
The action of travelling by train is repeated but it is not continuous.
In any case, the explanation for a person's noticeable weight gain could depend on multiple factors, not only eating too many pizzas. Barring serious metabolism and thyroid issues, the most common reason for putting on weight is plain and simple overeating and lack of exercise.
He must be eating too much, mustn't he? (a general fact)
He must have been eating a lot recently.
(a repeated action that began in the past and persists in the present)