Stainless steel is not completely non-magnetic as there will be ferrite in the steel. So it might be worth confirming the properties of a particular steel.
You know the stirrers you get in labs. You drop a bar magnet into a beaker and it rotates and stirs the contents. Is the drive for the magnet not simply provided by having another bar magnet rotated by motor underneath the beaker? Would that not be a simpler way of getting the stirring action, and an easily variable rotational speed?
Is a glass or ceramic container not an option?