This would only involve the BBC investing in 1(?) additional channel on satellite. This may be more difficult on terrestrial and cable.
Terrestrial is actually easier now they have the centralised mux operation. Usual configuration is Net1 distribution goes to the region and the output of their opt switch is fed to the mux centre where it goes into the multiplexer for the region and then on to the transmitters. It should be possible (as it's all done with switching matrices, probably computer controlled) to put Net1 directly into each muxer, put Net2 on the distribution circuits which go through the regions and put the return feeds into the BBC2 input on each muxer. This same principle could be used for any BBC channel.
For satellite it would be necessary to be able to switch all of the BBC1
EPG channels to a single stream (which isn't really viable). For the few occasions on which it's needed a red button app would be tolerable for the regional variations on BBC2 rather than buying loads of EPG slots.