tapas crawl!? did someone say tapas crawl??! We're in!
but only in Burgos???
Hi, traveler, if you're looking for sites worth visiting close to the Camino, accessible either by bus or taxi, you're going to find a LOT. Near Burgos is the monastery of Santo Domingo de Silos. If you search through the forum, you'll see that many have recommended a side trip there. Given the bus schedules, you're likely to need at least two nights there to be able to visit the church/cloister and attend vespers. With your schedule, that'll be very easy to accomplish.
If you like ancient churches, about 25 kms from Leon is the amazing San Miguel de la Escalada. Public transportation may be tough, but it is an absolutely beautiful mozarabic church.
Near Belorado is the monastery where I believe the Castilian language was first written, Yuso and Suso (can't remember which is the older one). I assume you could more easily get there from Santo Domingo de la Calzada, but we met a guy walking to Belorado who was out for his daily constitutional and he volunteered to pick us up in the afternoon after we had arrived and settled in the albergue and take us there in his car.
The excavations/remains at ATapuerca have also been described a lot in this forum, and many recommend a visit as a day trip from Burgos. These are just the ones that jump to mind, but I'm sure there are lots more that I'm not thinking of. With your timeframe, you will be able to take advantage of a lot of visits!
But I do echo Kinky's recommendation about making sure to save 3 or 4 days to be able to walk to Finisterre or Muxia (actually, the absolute best is to walk to Finisterre AND Muxia, so that would be 4 or 5 days!)
Buen camino, Laurie