I have stayed several times at the Academie. We have tended to use it as our last-night hotel before returning home, but at least once we stayed for several days.
The hotel is well maintained and attractive, if a bit heavy on elaborate period flourishes. Just be aware that the rooms are quite small, even for France, and one room didn’t even have a dresser, just a very small armoire with a few shelves. There was a small desk and chair and the bed, and that was it. Pictures on the website are carefully framed but you can see how close the foot of the bed is to the opposite wall.
Staff is friendly and accommodating.
It worked out well for us because we liked the location, a few blocks from the buzziest, tourist-driven parts of St Germain but a short walk to them nonetheless.
Not sure how it compares in terms of price, but we had reserved at the Hotel de Seine for part of a trip that we unfortunately had to cancel last year. We have not stayed there, but it’s quite popular and the decor is also classic French. Maybe worth a look.
Sorry, but I don’t know the Hotel Baume.