If you wanted to bring back more than 8-12 (or maybe only with a case) it may be cheaper to ship the wines back vs. check them.
You will save 20% on the price of the wine due to VAT. Depending how nice the bottles are, this can be quite a discount.
I would reach out to the wineries you plan to visit, and see if they can arrange it or if they have a shipping/import service they typically use.
I'm not sure if you're ability to do this will be impacted by the 3 areas. I'd ask.
If you can't do that - I would check to make sure you can't just buy these wines in the USA for less.
If you decide to check a bag, I would either use:
I've used both many times with no problem.