Hi, Nick,
You could conceivably attend both, but that would be a stretch, as you're flying in on the Thursday. They're not that far apart. Cowal is Thursday through Saturday; Birnam is Saturday only. I'm sure that some of the athletes compete in both. However, it all depends on how much athletics/pipe bands/Highland dancing you want to see and hear, and how many meat pies you want to eat!.
Cowal is huge. It's one of the biggest in the world, let alone Scotland. This means that you would pay a premium for lodging, if you can find any within about a 50 mile radius. Dunoon is on the edge of the Cowal Peninsula, and there aren't a lot of other population centers nearby where you'd have a large selection of lodging. Many people come over on the ferry from Gourock - it's possible to park in Gourock and take the ferry over for the day. That would open up a lot more lodging possibilities. The ferry runs frequently.
The Birnam Games are in a very pretty part of Scotland. As it's centrally located, you'd have a much better choice of accommodation. Stirling is fairly close by, and would provide a greater choice of lodging. Also, it's a smaller gathering, but as noted above, there could be a large carryover from those folks skipping the last day of Cowal.
Both Dunoon and Birnam are within a reasonable drive of Oban and Glencoe.
Personally, I'd plump for Birnam, just because it's smaller. It would be easy to get lost in the shuffle at Cowal. If you're a wee shortarse like me, it's easy to get stuck behind tall people in the crowds, and not be able to see anything! As you're flying in on the 28th, I think that Birnam would be your better choice, as you'd have missed the first day at Cowal.
I hope that whichever (or both!) Gathering you choose, you have a wonderful time in Scotland!
Best wishes,
Mike (Auchterless)