I would recommend Biergarten over San Angel Inn.
We enjoyed the atmosphere at San Angel Inn, but the food and the service weren't very good so we won't be going back for a while. We tried it twice just to make sure it wasn't an off night the first time and had similar experiences both times.
Biergarten was a very pleasant surprise for me. I went with a friend for the CP dinner package, and I was expecting to enjoy the atmosphere (which I did) but wasn't really expecting to find much to eat. I'm a vegetarian and don't like heavy foods. I was pleased to find several light, non-meat selections on the buffet. Everything I tried was really good. We have ADR's at Biergarten for our October trip.
I was nervous about being seated with other guests at Biergarten, but it turned out just fine. We had a family of four on one side of us. They were very friendly and we had a nice conversation with them throughout dinner. There was a young couple on the other side of us, and we could tell by their body language that they were just interested in talking to one another, but that was fine - it wasn't at all awkward having them seated with us but not making conversation with them. It really didn't feel any different than when someone is seated at another table near you in any restaurant. In fact, the tables at San Angel Inn are so close together, it was more of an issue bumping chairs with those dining around us there than having other guests seated with us at Biergarten.