Yup, you're going to need new wheels, 14" or 15". There aren't discs to clear 13" wheels (and your tire options open
way up if you go to 14" or 15" wheels).
A pivotal decision is whether to keep the existing 4" wheel bolt circle or move to the 4½" bolt circle. Wheel availability and brake options are both much narrower in the small bolt circle than in the large. That argues strongly in favour of going to the large bolt circle, especially since you'll need new wheels anyhow, though it does mean you'll have to do something about the rear axle which presently requires wheels with the 4" bolt circle. There are no drop-in axle shafts with the larger bolt circle; the usual course of action is to swap in a later rear axle assembly—usually the stronger 8¼" unit to replace the original 7¼", in the process also getting larger 10" rear brakes. So there's a lot more to think about than just what kind of brakes you want; there will be additional effort and expense to do a clean job of it (pick brakes that are easy and inexpensive to service, have one kind of wheel on all four corners so you're not messing around with two spare tires and wheels, etc.).
There is only one factory disc setup with the small bolt circle, a Kelsey-Hayes 4-piston setup used on A-body Mopars from '66-'72 and almost identical to that used on '66-'68ish Mustangs. When in good condition, it works well, but parts availability is on-again/off-again spotty. I have this setup on my '62 Lancer ('66-'72 disc brakes with factory pads and no power booster), and I like it, but I'm not sure I'd choose it as the upgrade if I were starting again.
The A-body got a single-piston floating-caliper disc brake for '73, together with the beefier ball joints and larger wheel bolt circle the bigger Mopars had been using for years. This '73 type setup was used on rear-drive Mopars clear up through 1989, meaning system parts are widely available and inexpensive. I really like this setup in my '73 Dart (big-bolt discs w/factory power booster, upgraded to '76-up calipers, which have a larger piston/bore, with Bendix metallic pads). Changeover of your '66 to this system using later factory parts (some good used, some new or properly rebuilt) is not very difficult or costly if you jump on an ad like
this one, and leaves room in the budget for
good tires on nicer wheels.
A '66-'72 type 4-piston disc setup is available in either bolt circle from Stainless Steel Brakes Co. You can either spend $1100 for OE type components (cast iron calipers, etc.) or $734 for what look to be nicely machined aluminum calipers; I am looking at
this page. All premium components, to be sure. But we're already well past your budget limit if we're buying one of these off-the-shelf brake changeover kits, because we also need new hydraulic lines and valves and a master cylinder and wheels and tires.
If you have no objection to using non-Chrysler parts on your car (and you shouldn't, for it's possible to make great improvements inexpensively by paying attention and adapting the
good engineering where it can be done, no matter what company put it out; for example the
HEI upgrade), there's the
Scarebird option, which uses Chev rotors and Toyota calipers together with apposite bracketry to replace the drums with an inexpensive-to-service disc setup, and without changing the wheel bolt circle—but here again, you'll still have to have 14" or larger wheels, and wheel options are much greater in the large bolt circle. I think the Scarebird setup can be had in the large bolt circle, too, but I'm not certain; check with them.
Master cylinders, booster-or-not, hydraulics, etc. are all easy to spec out once you pick what kind of brakes you want.
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