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.
Correct. Just screw a grease gun hose into the fluid port and pump away. If it's a dual piston calliper, you will need to put whichever one pops out first back in, clamp it in place an repeat for piston number two. Even tightly seized pistons come out quite easily.

On permit aircraft we get the calliper grit-blasted clean of all paint and grot, and dye-check for cracks. Then simply skim out the bore in the calliper, make a new piston (it's only a very simple aluminium puck) that is the same amount oversize as the amount we have opened-up the bore. The o-ring groove is also bigger by the same increment. Assemble with new seals (same as the original), bleed, test and fly.

Using the same size seal negates any chance of a future owner failing to read the logbooks and not using the right seal (as might happen if the seal size had changed).

No need for any involvement from the LAA. It's a simple repair well within the scope of inspector sign-off. I have no idea if it is possible for a LAE to authorise a simple repair like this to CoA aircraft as I only work on Permit aeroplanes...

Failure to maintain brakes properly is a classic 'cost saving' gotcha. The cost of keeping the brakes lubricated and the fluid changed every couple of years is way less than replacing the brakes when they go wrong (not to mention the cost and trouble pulling the aircraft out of a hedge...). But you'd be surprised how often brakes are neglected...
One of the problems is that Cleveland appointed Boeing/Aviall as the Sole Master Distributor for their product line.

The smaller/medium sized parts distributors who offer service and more local support have been shut out of the market and the product is under monopoly control for after-market support and sales. In the last two years the price of Cleveland wheels and brake parts has skyrocketed. It appears the management at Cleveland are only driven by annual results and have "sold out" their brand to a third party. There are more and more PMAs being approved. I use a PMA or alternative whenever I can. Such a shame when a great product and brand ends up being destroyed by "here today gone tomorrow " management with such short term planning and the cost is just passed on to the end user.

I have voted with my feet.