Miata pistons will work just fine, there is absolutely no problem with using them, so long as they are stock compression for your model.
Heres the Compression Ratio for each Model
NA Miata: 9.0:1
Base Naturally Aspirated Model Capri with Manual Gearbox: 9.4:1
Base Naturally Aspirated Model Capri with Automatic Gearbox 9.0:1
XR-2 Turbocharged Capri with Manual Gearbox: 7.9:1
The bore on the all of these pistons stock from factory is 78mm
The problem is your going to have a hard time finding a .030 bore, right now its very easy to find aftermarket pistons in .010, and .020 sizes both of which are available from Weisco.
What they have done is a +.75mm bore, and with this engine there isn't a large water-jacket, so even if you could find pistons(which you can have Weisco custom make for you), you could have problems with needing to sleeve the engine cylinder due to the jacket being so close.
Also the machine shop is in the right here, much/all of these old parts need to be replaced, you cant just put in old rod bearings, and piston rings and the rest back in and call it good when you put new pistons in. All of these components wear just like your brakes do, and in this case they can be completely ruined when the crank pulley snapped.
You absolutely need to rebuild the whole engine if you want it to be reliable, I realize this is not a cheap option, but its a much better option then buying a used engine from another Capri. Another thing you need to make sure is if you will go down the route of rebuilding this engine is you need to make sure you have a long nose crank on the new engine, if you don't you can have this happen again down the road, there are fixes involving locktite, but its unlikely the engine builder will be willing to do them for you. Overall the Machine shop is trying to save themselves and you the problem of the car breaking down in 100 miles, because they did things the wrong way. It may seem like they are bullying you but that is not the case at all, everything they want to do inside that engine is reasonable. The only thing they are wrong about is that the Miata pistons work just fine. What compression ratio was the pistons you asked them about? if they are higher then stock 9.4:1 you can and will have problems with leaning out the stock ECU.
Another thing when you do this is your going to want to get the head checked, resurfaced(if needed), replace the oil-pump, water-pump and get new seals throughout. I would also add I would replace the throw-out bearing, and have the flywheel replaced of resurfaced, and a new clutch put in. If you do all of this you are likely to have less issues down the road.