The place for technical discussions about GA and flying.
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Judging by @Charliesixtysix 's photos, I'd guess that the body is Aluminium Alloy, (or galvanised (zinc-plated ) steel. The piston is probably hard-chromed steel. I'd suggest Phosphoric acid dribbled down the gap 'twixt body and piston. It converts rust into ferrous phosphate, which is pretty-well inert (a black "crust" )

You should be able to make a "retaining-wall" from plasticine and just fill the piston-area with a bath of Phosphoric acid (Jenolite, some alloy wheel- cleaners, -even Coca -Cola ! -I kid you not! "Coke" has a fair bit of Phos. acid in it and that's why it rots teeth! buy a large bottle, get a plastic food-container and just pickle the whole caliper for a day or two. An old automotive brake-or-clutch master -cylinder could be connected to the caliper and non-fizzy fluid used to force out the pistons then it's just clean and reseal -unless the seal-grooves need re-machining..

Try leaving a bit of rusty metal in Coke, you'll be amazed at the result.
I recently tried a bio-friendly product called "Fertan" which is Tannic-Acid based. was used successfully to reclaim some 1950's Rolls Royce Cloud wheel-cylinders. black residue brushed off.
^^^^^^^ potentially explosive release! OK if you are practised in the technique, but otherwise hydraulic pressure is so much more controllable (and a higher force is so much more controllable and easier. (all gas-cylinders are tested that way)

Forgot to mention sticking a caliper in the oven/cooking with a fan-heater.......even better if the piston is accessible for a squirt of freeze-spray when the caliper's hot . (differential expansion)
Another good rust remover is borax. It dissolves the rust and leaves the surface clean but pitted. Used on the old, old bike that had sat in the brambles for 30 years. Deox-C is one brand name but you can get the raw material easy enough.

Now - I have in the past repaired calipers using chemical metal squeegeed into the walls with a Stanley blade. Once set, a quick sand and hone and you are back to a new caliper. I qualify this by saying the caliper was probably impossible to source easily.

Original question asked which brakes I would use - Grove every time. family run, great to work with on special projects, we designed and had made a new undercarriage for a Brandli Cherry. Robbie is a brilliant Engineer and his gun drilled RV8 undercarriage legs are a work of art.