If you do have tea at the Jane Austen Centre really, don't waste your money on a tour. It doesn't have any of Jane's things but has pictures of writings, clothes, etc. so is pretty disappointing. If those in your group are Jane fans they will probably know more than is presented in the Centre. She never lived in this house altho she might have walked by here on the sidewalk as she lived in nearby Queen Square for a time.
If the Centre is offering walking tours when you are there I do highly recommend them. They are excellent.
You should go in the Pump Room even if you don't have tea there. You can get a free drink of mineral water served over by the far windows overlooking the baths.
If your travel time coincides with the Jane Austen Festival definitely do some of the activities. It was such fun seeing people walk thru the city wearing period costumes.
http://www.janeaustenfestivalbath.co.uk/
I'm not one for Tea (vegan so usually can't eat any of the sandwiches or sweets) but for ambiance and being in the room all those Regency novels visit I'd vote for the Pump Room.