The summer season is the busiest in Batumi. The Black Sea beaches draw a lot of tourists to the vibrant city of Batumi. But for birders, the high season is in autumn. From mid-August well into October, any multiple-day visit will result in good birding. In good weather, one of the watchpoints will result in the visible migration of raptors, storks, and other birds. If you are keen to witness one of the peak days, then you should time your visit to include the first days of September or the first days of October. The former is classically the peak migration of Honey-buzzard and Harriers (but it can be different with a couple of days), the latter is good for the steppe buzzard and eagle migration.
Spring can be very rewarding as well. A visit between the beginning of April and the end of May should result in stunning raptor migration, although numbers are not that high and constant as in autumn. You are likely to be treated with a nice diversity on a good day. Also, the migration through the area is enormous, and birds are at their finest, in breeding plumage. Imagine a fall of Black-headed buntings.
Winter birding should not be underestimated, as the southern Black-sea shores and wetlands are an important wintering area for waterfowl, divers, and gulls.