I say go with the ring because it will go with everything, while the Birkin would only go with certain outfits/colors depending on what you get.
Other than that, I'm sick over the investment options. Overpriced designer good #1 or overprice designer good #2. At least the Birkin would be a recognizable item. The ring may as well be from Kay Jewelers for all anyone else would know. I second the opinions that a diamond of equal quality can be had elsewhere for much less.
ITA - with everything you said.
Truth is, OP - no one is going to know your ring is from Tiffany's just by looking at your hand - and it does sound like you are going for "recognition". So - with that in mind, I say go Hermes.