So the gearbox is outside and below the oven?

Why not just run the shaft through a coolant bath with the gearbox below it?

Something large enough to dissipate the heat.

I don't think it would have to be fancy...................just circulate water through it.

I was thinking of a helical spring-like coupler between the two sections of the shaft. This would be similar to the spring-like handles used for skillets used on old wood-burning cook stoves and the tools used to move the metal covers on the cook stove surface. These handles stayed cool even though the stove top could be 400+ degrees F.

Like this, only much stiffer and probably just cylindrical, not bi-conical:

I think I may have it.

Run your metal shaft out of the gearbox. Have a flange on the upper end. Make the rest of the shaft that goes into the oven out of ceramic.

The ceramic shaft can be flanged at both ends to mate up with the metal components. Then all you need is a high temp gasket between the mating surfaces.

Now you just need to find a suitable ceramic shaft................or have one made.

This came up when I googled high temp ceramic shafts.

Or you could use a ceramic coupling between the furnace shaft and the gearbox shaft,

A little cunning design could ensure that the ceramic coupling is purely a compression member that never sees either tension or torque - something like a dog clutch with ceramic spacers between the dogs.

A magnetic coupling should not transfer heat, a little fan blowing on the gap would help.

There are mechanical couplings that have a piece of spring steel weaving between the teeth of the flanges. They come with a cover to keep grease in, but at your speed that would not be necessary.

There are couplings that use a piece of three-tab rubber to transfer the load. Make one out of insulator material instead. Bakelite is my first thought.

As a matter of fact, look for bakelite couplings! Strong and insulating.

A chain drive with coolant can be used.

If this is for centrifugal casting of mirror blanks, with stress relief on cooldown, then the spring will not work - too wobbly, you will need some sort of flywheel mass

Large diameter aluminum plate on the coupler.

I have had the occasion to use steam as a coolant for a 600 degree F hot oil pump seal. If you have low pressure steam available in the area, you might be able to use it as a "once through" fluid through a stuffing box or shaft seal enclosure to cool off the drive shaft below the oven.

It sounds weird, but compared to the 1800F furnace, steam at 250F is relatively cool! And because the steam is relatively clean, it helped me extend pump repair requirements from 3 months to a 12 month interval.

Graphite is used to stir molen metal and is ideal for high tempature applications. The shafts are graphite also. I don't know what the bearings are made of.

Lovejoy manufactures a "spider" type of shaft coupler. One of the compounds available to join the two halfs is made from Hytrel. This product will stand up to 250 degrees. The heat flinger disc should help. You could always mist the spinning disc with water to disapate additional heat. Good luck.

How about a ceramic coupling? Google "ceramic couplings" and choose the surplussales.com website. They have quite a selection!

The loads are very modest. So are the speeds. I'll simply run the shafts at 90 degrees into a sheet metal water bath and use 45 degree bevel gears inside. A simple float will keep the level proper.

Thanks to all who supported this inquiry.