A few immediate options I can think of:
1. Book a hotel to work from for the day - dayuse.co.uk
If work are paying, the Mercure on Welshback in the centre is very nice and quiet, as is the Premier Inn on Queen's Square, but a bit cheaper. Ibis is new, next to Bristol Temple Meads (2 minute walk) and cheap, but zero to do around there in the evening.
2. Cafe Revival if you want a decent breakfast after your first meeting: caferevival.co.uk
If you want to work somewhere quiet whilst you're having breakfast, order your food then head back up the stairs to the top floor. It will get busy but you can get some sensitive work done here without being overlooked if the sofa's still free.
3. The Old Fish Market pub has some quiet nooks and crannies. oldfishmarket.co.uk
4. Consider The Watershed - next to the harbour, go upstairs to the cafe, lots of places to work, some public, some less so, including a balcony if you're hardy. Food and coffee available too. Trendy, so gets busy quickly.
5. Bristol Library: College Green, Bristol BS1 5TL. Reading and working spaces, not sure if you need to join to use the private ones but you should be able to at least get in to work.
6. There's also a large meeting area in Colston Hall (renamed the Bristol Beacon last month). Might be too public for your work, or not even open at the moment, but it has a drinks/food bar too on the ground floor: Trenchard St, Bristol BS1 5AR
7. Finally, check out the Arnolfini - gets busy but might be OK for you to work: arnolfini.org.uk 16 Narrow Quay, Bristol BS1 4QA
Mostly an arts centre, but the cafe was decent last time I was there.
Failing this, keep an eye out for the loads of small cafes and pubs as you walk around.
So long as you're consuming something slowly you can probably stay in any of them all day, but it's easy to move from one to another with a short stroll, plus you get to see a bit of Bristol.
All of these are within a 10 minute walk of each other, and in central Bristol, approx 15 minutes easy walk from Bristol Temple Meads.
Postcodes are all BS1, so you should be able to find some entertainment close by too ;-)
There's plenty on here, but I'll let someone else have a go at answering as I'm not too au fait with SPs in the area.
Hope this is helpful.