The majority of BX owners, including myself, are of the opinion that R4s cause little turf damage under proper conditions, and provide good traction when doing loader or snow work, while turfs do not provide the necessary traction for snow or loader work. I cannot speak with experience about the B series.
I can say that the only time my R4s have had problems on the turf is when the clay soil has been supersaturated by too much rain or irrigation, or when I forgot and left the tractor in 4wd. When turf is not overly wet, and I am in 2wd, I never have problems. (Were you in 4wd? If so, you might go do a couple of turns on the turf in 2wd and see if it differs.)
When I bought the tractor, I got both sets of tires on separate rims. I also bought a hydraulic jack, wheel chocks, and electric impact wrench, all on sale and cheap at Harbor Freight. This allowed me to change tires NASCAR pit stop quickly (actually, more like 4 min.) whenever I thought I needed to change tires for a different application. Thing is, I found I couldn't do loader work with the turfs on, and the R4s didn't harm the sod, so the turfs have been laying under the house for almost 6 years.
The experiences of others may differ from mine so consider the advice that best rings true to your conditions. Good luck with your decision.