My guess, and it's a pure guess, is that you might have been able to order dual exhaust on a J code 360 SE or have it installed by the dealer. There are part numbers for dual exhaust components for 4 door B-bodies and B-body station wagons as well as P, D, and C bodies with engines ranging from 318's to 440's including the 360(and I presume the 2 and 4 bbl 360's although there is no distinction between the two in the parts manual). I also found the "Flange, Exhaust Pipe Ball" listed for "w/360Eng.,w/o/HPerf." and it is the same PN as for the High Performance 360. The only other instances where I see dual exhaust components referred to for 360's other than the high performance 360 are for clamps for the exhaust pipes and tail pipes and tail pipe support hardware. Some of these components are listed separately from the B-body 4 door and station wagon dual exhaust, but not all. It's possible they were only intended for the 4 door and station wagon dual exhaust options. Again the only way to know for sure it was possible would be to find dual exhaust called out in a build sheet for an SE or something in the sales literature indicating dual exhaust could be ordered on an SE.
It looks like from 69CoronetRT's post above that the 4 bbl 360 SE would have come with N95, emission controls, which meant California emissions controls. My Road Runner is a California car and does have the N95 Emission control code on the build sheet. Correct me if I am wrong, but I think sometimes the N95 code also required the N97 noise reduction package which came with tail pipe turn downs instead of chrome tips. My ROAD Runner did come with chrome tips even though it has the N95 emissions code so I do not know if my memory is serving me well.
Finally, if it were me and the car is not original I'd put dual exhaust on it and tell the naysayers that is the way I'd have wanted to be able to order it in the first place.