One of the problems here (and I don't know why Cal doesn't solve it) is brand name. Cal cannot decide whether it is Cal, or UC Berkeley, or Berkeley or Cal-Berkeley or whatever. That lessens the academic prestige that admitting "Cal" would give to the Big Ten. Here in the east, people don't know whether Cal is Berkeley or Cal Tech or what.
I think part of the ingrained academic vs. athletic tension seems to be further propagated by each side having its own brand. Seems intentional albeit shooting yourself in the foot in the overall scheme. Difficult to get folks to pull together and go in the same direction.
Though it's not impossible to thrive with both an athletic brand and an academic brand, I think one of them has to trump the other in terms of the outside world (beyond the campus walls) and general public at-large or else the messaging becomes convoluted.
The problem may be that the academic brand of Berkeley is held in such high esteem the world over that it overshadows the athletic brand of Cal, whereas the latter is the one that is typically leveraged for marketing purposes as it potentially helps bring more eyeballs and folks to the front doorstep of the university. The conundrum apparently lies in that more people might be drawn to Berkeley due to the outstanding academic reputation versus just regular folks being drawn from having seen the sports teams on TV to then learn about the school.
It would appear from afar that Berkeley doesn't really need to lean on Cal's revenue generating sports such as football/hoops to bring more visibility to its academic programs, it already has that in spades, even if that's mostly a cross section of those already inclined to be more interested in academic and research endeavors.