The octave lifter rod on the sax body at the top broke on my Buescher alto...
The rod is probably different than the other brands that you mentioned. I'm assuming you mean the floating lever rod that attaches this key to the octave lever. Aside from the problem of finding a new rod/arbor/hinge screw... typically these screws don't just break. They break because there is some kind of stress or wear or damage... or on occasion it comes loose and is lost. Replacing the screw may not fix the total problem. As a repair tech, most of us provide much more than the part replacement. If the customer asked me to just replace the screw, I would feel obligated to repair the problems that made the screw break and the mechanism stop working. Typically in an octave mechanism it is because something has been bent multiple times and it has broken due to the stress. So going to a tech to replace the screw for a fairly nominal price would probably be less expensive in the long run if you don't have the capability to replace the screw and fix what ever problem made it break.
FYI, making screws is a pretty routine venture for most repair people. I have a stock of drill rod I make screws from. The typical process is to start by accurately measuring the old screw then select a drill rod that will slide through the key. I then check the next larger size(s) to see if the key is worn and will accept a larger size rod. If a rod will slide partially through the key then I know that the key is bent. The next step is to straighten the key and re check the rod. The last step is to select the proper die to make the threads and thread the end of the rod. A while back there was a lengthy and heated discussion on how to do this so you can probably search the site for the many different and successful methods for cutting screw threads. Oh! and lastly if the rod is only bent and not broken as you said, it can be straightened in about two minutes most of the time and may not need to be replaced.
Shorter answer

No, I don't have a screw, No the other brands probably won't work, yes I can make one and so can your local repairman most likely.