Amy, what a wonderful trip and you are giving it enough time. In general I will say that your days have a lot of free time at the end. That’s not an issue, as there is a lot to see, do, relax and enjoy.
Wednesday, 2 Sept: Castle Hill/Mathias Church/Várkert Bazár/Hospital in the Rock and Nuclear Bunker Museum: A good use of half a day, and arrival day. But if you care arriving from the US you might want to do a lower key day, but that depends on where you are staying.
Thursday, 3 Sept: Hungarian Parliament (get your tickets now if not already)/Great Synagogue/Lunch in the Jewish Quarter/Great Market Hall/Walk across Chain Bridge/Danube River cruise: Are you staying in Buda? Is that why you list walking across the bridge? If you are staying in Pest, then you did that the day prior. Otherwise, all your stops line up nice along the 2 Tram and the 47/49 Tram. With a starting location (hotel) someone could comment on the route and order and some other stops along the way; for instance, there are some great antique print and book shops if you are moving in one direction and a labyrinth of connecting basements full of antiques if you are going the other direction.
For your lunch in the Jewish quarter if you want to touch the culture in a unique way, look for a Kosher restaurant. Here are a few to look at:
Friday, 4 Sept: Heroes' Square/City Park/Vajdahunyad Castle/Széchenyi Thermal baths
For City Park there is the House of Music and the Ethnographic Museum both to tour or just for the architecture. The outdoor train museum isn’t far. There are some good restaurants close by. Good meaning either great food or interesting experiences. You will be done by mid afternoon even if you take a long soak at the baths, so again, a starting/ending point (hotel) help with suggestions.
This is a Friday, a great night for the theater. Opera, Oprett, Musical …. Several great venues in town. But for the Opera, buy now.
For our first few days (just the two of us), we’d love to do something
a bit special for our anniversary. We really enjoy wine and were
thinking about heading to Eger and checking out Szépasszony-völgy.
Eger is always a good idea and for Szépasszony-völgy hire a guide for the day. See if The Senator Haz still has rooms available. If not look on google maps of the location and find something that meets your criteria that is close to The Senator Haz. Balaton, not unless romantic to you is a modern resort hotel. Tokaj is more of a trip out of Eger same guide, same day.
Another suggestion is there are some wine tour companies that do overnights at wineries out of Budapest. Some ae very close and some a bit further. I also know a young lady that does these bespoke if she is available. I will send you contact information. I love these sorts fo things and the wine.
