Peak season should be Nov - Jan, Feb should be fine.
Why go Phuket, so expensive place?
Since you will be in BKK, can consider Koh Chang and Koh Kut. Both are ulu island so beach and island remain beautiful.
Like to go phuket during jul to sep ...
hotel very cheap ... 3* about 20+ only
if is nov to jan is about 50+
air ticket: phuket usually below 100, bkk usually 100 to 130 if promo
hotel: about the same price for both on average (unless is 5* or villa)
airport transfer: 600 (1-hr phuket), 400+ (35min bkk)
transportation: phuket very expensive for both taxi and grab
