Does anyone have a reliable method of metering caustic soda flakes into a tank of water? The hygroscopic nature leads me away from rotary valves and screw feeders.

Well about 30 odd years ago ICI plastics used vibrating feeders to slowly feed solids into water as part of PVC. Not the easiest things to maintain though.

If I remember correctly they were Vibromatic units

What method is used to feed caustic flakes into the tank. The method will depend upon that.

The hygroscopic nature leads to the caustic gradually losing its potency.
Why not use a screw feeder or rotary valve fed by a lidded hopper. A modest N2 purge will keep air out. Purge it at feeder level and lead the vent to outdoors, or ensure good ventilation.

Or use a heatless dryer to dry air. Use the dry air to purge the hopper

Have you considered using a density meter? If you are making a batch of caustic you can circulate until the desired density it met, stop the conveying plant, empty the tank then start again. Nucleonic density meters are quite common in the sugar industry but have legal issues if you have no other radioactive sources on site. A MicroMotion density meter are also a good choice.

You can use hydraulic ram feeders.

metering will be based on cross section of feeder barrel,stroke of feeder ram and density of caustic flakes.
If you so desire I can send you a sketch of the feeder that would do the functions you desire, effectivey.