I'm replaceing a portion of drywall on the ceiling between floors 1 and 2. For the drywall, I plan to add backer boards on edges. The issue is there are joist blocker boards running perpendicular between the joists. Can I "legally" notch a quarter inch within the center one-third of the perpendicular joist blocker board?
You are adding nailing edges to the edges of the existing drywall to provide support for the edge of the new drywall sheet and the edge of the existing. A lot of people will just get some 1x4 and span the old sheet to the new sheet. The two sheets then align at the joint. They might not be "sucked up" as much as if you adding a nailing edge that was connected to the joists but sheets of drywall are typically supported 16" on center so at most the existing sheet will have support < 16" away and the new sheet will have support < 16" away. The 1x4 between the sheets then lends support from one sheet to the other. These pieces don't even have to be continuous. You also have texture on the ceiling which works to hide all kinds of defects so your work can be even less precise.
As far as your question goes, you can notch the blocking between the joists. The solid blocking between the joists allows your floor assembly to work as a single unit. Often cross bridging is installed instead of solid blocking and they have comparable performance. Blocking is also routinely knocked out by plumbers. A notch to the bottom of the blocking is not going to effect the function. The bottom of the blocking is not being loaded in tension. If you are worried about it