What metal can be permanently magnetized and remain stainless and workable as small washers

A. I do not think that there is any metal that will reach the Utopian status that you are seeking.
First, these are very small, so they will not have a large magnetic force. Actually, it will be rather small.
The 400 series SS can be magnetized, but I would not call it permanent, by my definition, and it will rust in many cases.
I would try AlNiCo which is a common magnet. True some are used in super electromagnets, but it might be worth a try.

A. All metals are subject to oxidation when exposed to humidity. Even stainless steel. I think alternatives would be better.

You could use a magnetizable ceramic. For example, you could make a slurry with magnetite and binders, and bake these washers.

Another option would be to use a ferromagnetic metal, and coat it or passivate it with a corrosion-resistant material with a good permeability. This is usually what's done with rare earth metals.

A. I have spent quite a long time reading about magnets as they massively affect the sound of any musical pickups that rely on them or speakers that use them in the drive coils. I was also interested in them due to their use in magnetic suspension in things like turbo molecular pumps. But I can't profess to know everything possible with them.

I can't think of any strongly magnetic materials that are also stainless. One of the definitions of stainless is that it's not magnetic.

That brings me to a suggestion that you use something like neo-dymium magnetics (super magnets) and coat them. There are a bunch of places selling them online and I've seen some washer like versions. Whether or not they'll be as thin as you want them I'm not sure.

Also, a lot of magnetic materials don't like being magnetised along their short dimension, e.g. the thickness of a washer.

In terms of coatings, there a whole lot of them at your disposal. For medical use, a good thick coat of chrome would be a starting point.

From there, I'd go for more exotic coatings like one of the platinum group; platinum or rhodium maybe.

Other ideas could be an epoxy powder coat finish or some form of teflon. DuPont do a liquid teflon that can be applied and then dried onto the surface. PTFE is of coarse a big player in medical instruments since it'll go through an autoclave and is all but 100% inert.

A. Firstly, no stainless steel is totally corrosion resistant, so there must be a compromise between materials and its operating environment. Secondly, only 200 and 400 series stainless steels are magnetic, although 300 series will become slightly magnetic under some circumstances. You may find the 200 series offers you the best option (201, 202 perhaps).
Other metals, such as nickel and cobalt will exhibit a degree of magnetism and corrosion resistance, but the level of resistance is dependent on the environment. You could also perhaps use a high nickel - chrome alloy that will have a degree of magnetism, but not as much as a stainless steel - this will be more corrosion resilient than stainless steel, but again, nothing can be certain for your environment.
You could also talk to the University of Birmingham Materials Science or Metallurgy departments - they may be able to help you out; they are based in Edgbaston.