Its seems likely to have been an extremely costly and difficult task.

Firstly in terms of the naval battle that'd have to be fought. The Germans would have had to rely primarily on the Bulgarian and Romanian navies in addition to what ships it could transport over land. The 1936 Montreux Convention and Turkish neutrality in the war meant that the Germans (and allies) weren't able to transport naval vessels through the Bosphorus and Dardanelles.

This meant the Axis forces were mostly relying on a handful of destroyers in addition to submarines and torpedo boats. The Germans and their other axis allies were able to wage a harassing war against Soviet naval forces for a few years historically in the black sea. But had come to the kind of decisive fleet engagement a major naval landing would likely require the Soviet naval tonnage would have been a pretty major advantage in a fleet engagement, with the Axis forces lacking Battleships or cruisers to match the Soviets.

Committing sufficient air power might have increased the chances of success. Taking off from Romanian to fight somewhere around Odessa or Sevastopol (if we're talking about a 1941 engagement) could have been viable. But particularly in 1941 had the ultimate objective been Georgia they'd not have a base that'd put a lot of their single engine aircraft in range. That's your Ju-87 for dive bombing or most part 1941 Bf 109 and Fockes which would be out of aa comfortable combat range. So you'd end up having to exclusively twin engine aircraft like like t Bf 110 for your fighter support and things like Ju-88A-17s and He 111 H-5s and C series for torpedo and fighter bomber support. Certainly during the day Soviet fighters seem likely to have been able to inflict not insignificant casualties on this air support.

If we're waiting until mid 1942 when the Germans had successfully taken Sevastopol that does mean the Germans could have more easily based their aircraft in range. However you also have to bare in mind mid 1942 the Germans didn't have a huge amount of resources to be spending on opening costly new fronts. This might be instead of Case Blue or what have you but all that land still have to be holding the same ground in the Ukraine while trying to fight their way through the Georgia, up over the Likhi Range to get back down into the eastern Kur-Araz Lowland.

In addition to this Naval landings in general are hard. The Germans had had some success in Norway but they'd got pretty lucky, arguably there were several points of failure where that invasion could very easily have gone battle wrong. Now in Georgia the task would have been to attempt to successfully land and seize a suitable port city like Poti or Sukhumi as quickly as possible in order to successfully begin bringing supplies along with heavier equipment in. All while having to fight at the edge of their air range, against the Soviet black sea fleet at sea and against soviet land forces.