I'm always a little wary of weighing in on lubrication questions, because it can sometimes be really problematic using the wrong lubricant, and other times it doesn't matter so much.
There is an entire science to lubricant manufacture and selection, but much of it relates to industrial applications where machines are running continuously or for long periods under sometimes extreme conditions or at very high speeds. In these cases the right lubricant is pretty essential.
For things like bicycles you could use almost anything as long as it vaguely fits the application specs.
Movie camera lubrication is sort of in between, the tolerances are fine so you want to minimise wear, but the speeds are relatively slow, the materials fairly standard and the temperature range within normal climatic variations. When cameras were being professionally used and running through hundreds of thousands of feet of film a year the specified lubricants would certainly have saved the cameras from early deterioration and expensive repairs, but these days for owners who might run a few thousand feet through now and then the choice of lubricant is far less vital, as long as it meets the basic requirements.
With cameras of the vintage of a BL3 from the early 80s you're more likely to have electronic failures than mechanical ones, unless the camera becomes contaminated or subjected to extreme temperatures. The important thing is not to use lubricants that might react badly with certain metals, plastics or resins, or leach onto surfaces that require a specific friction or come in contact with the film or the optics. Avoid silicon at all costs for example, or consumer products like WD-40.
You could try to contact Arri about specified lubricants, I've bought Chronosynth 1/8 oil and Isoflex LDS 18/05 from them in the last few years. Or visit a rental house with an established service department (like the one I work for, I still have a small jar of Slick 50 kicking around).
Otherwise a decent clock oil like Nye Oil will suffice, and Arri's regular grease Isoflex LDS from Klueber would be fine for bearings and cams. Or ask a lubricant company to recommend something for the application. An important consideration is something that lasts a long time without drying out or changing characteristics. You don't need high temperature or high pressure greases.
One of the problems with changing greases is that mixing them can sometimes cause an unwanted reaction, so ideally you would clean off all of the old grease before applying the new grease. But that usually requires complete disassembly and the associated knowledge and equipment to reassemble and calibrate correctly. It's another reason I'm a bit hesitant to recommend alternatives, but the reality is that many people will try to maintain their cameras themselves anyway these days.