That cab is probably too small for the drivers in it, which is usually going to cause a low mid hump. They probably did that on purpose back then, because it makes the cab louder, but also sound kind of like crap for most things. A more serious BASS cab that's better designed for bass would be a better choice ("vintage" is fine, but maybe not QUITE so vintage, even an ACC 106 would probably be a lot better). The guitar amp probably isn't helping.
You could try using a parametric EQ and dial out maybe around 100 Hz or so. The problem is the reason the cab sounds like it does is because it can't really handle true low end, so as you cut 100 Hz (or whatever the offending frequency as, as discovered by trial and error sweeping that low cut) and bring up the overall volume to compensate, you start really hammering the cab lower down where it really can't hang. If you were playing mild mannered music in smaller places, that might not be an issue. But for loud and raucous and uncontrolled, you really want the cab to be way beyond what you need otherwise you'll probably be killing drivers on a regular basis.
You could pair the EQ with a HPF, and that will help with speaker preservation, but if you still need a sizeable low cut above that, you may find the sound is too thinned out to be usable.
As a start, to figure out the best course of action, I would probably try the cab with whatever you have that is closest to a traditional modern bass head. Something that you know "this is supposed to sound like a bass amp". If that alleviates your issues, then swapping out the head or adding additional EQ to the guitar head might get you where you want to be. But my best guess is that that cab is not really suited to what you are trying to do. It might be cool in its own way for the things that it's good at, but it could be a situation like "why doesn't the load of plywood fit in the back of my old MG?" or "why can't this 20 foot box truck corner like my old Lotus?" etc.... You have to use the right tools for the job.