How do I clean a vintage carbon steel pastry cutter?

Look up what has been written on restoring carbon steel knives, similar techniques apply.

To clean up loose rust, steel wool, sandpaper and wire brushes will help; afterwards, you might want to try and build a patina by treating it with a food-safe acid (vinegar, lemon...).

So for a liquid non-toxic cleaner go with Enviro Care Washroom Cleaner. (I know right!?!) It's made in the US.

Pastry cutters are odd shaped and have hard to get at spots. Soaking is better than spraying to get right into those spots.

Soapy water and a Scotch Brite pad (lightly) is useful after the soak. Rinse and dry thoroughly.

Mechanical abrasion, using sandpaper or steel wool or wire brushes, should work to get most of the rust off the surface of the pastry cutter. rackandboneman's answer has already mentioned this, I confess.

After the rust, or most of the rust, has been removed - you might try seasoning the pastry cutter, to prevent it from rusting in the future. You should be able to build up an oil-based seasoning, brushing on thin layers and heating to form thin layers of seasoning. You might be able to build up a tea-based seasoning, soaking it in thickly brewed boiling tea and letting dry, again several times, to build up layers of tannins deposited on the surface. I expect your choice might have to do with what sorts of products you expect the pastry cutter to be used with, and which you wouldn't mind wearing off into your pastry.

Both techniques are usually used to keep cast iron from rusting (the oil-based usually for frying pans, the tea-based for teapots). But, I see no reason that they can't be used to form a protective coating for your carbon steel pastry cutter to prevent rusting, if you're uncertain about rackandboneman's suggestion of using acid to build a patina with acid.

I would think that the pastry cutter should rarely need re-seasoning, I don't imagine it will get a lot of really hard use of the sort that will wear the seasoning away, but you can of course repeat the seasoning process whenever needed.