You can still get most older Colortran parts from Leviton/Colortran - at least I still do on 20 year old cyc lights. In fact I get these parts from my electrical supply house instead of thru a theater supply. This given my electrical supply has a sales person specifically dealing with the entertainment end.
I admit dealing with Colortran/Leviton/NSI since the buy out is a little more difficult or at least complex for vendors. The place I work for instance has had an account with NSI since before the buy out. While it might have at one point meant I could get Colortran parts due to this account, much less I could buy Leviton direct now, it's not the case any longer. This as opposed to the Advanced Devices (Bates) stage pinplug account once bought out by Mancuro enables me to buy direct the Mancuro line. This often is the case with other buy outs, but in the case with the Leviton one, it is complex.
If you are having problems in having tried this electrical supply and theater supply means or at least the sales people you have dealt with in putting the proper amount of effort in failed, TMB would probably be the best place to contact as they are in your area and in many ways specialize in solving problems like this. Believe I have also gotten this brand of part thru them.
What parts specifically are you trying to replace?
Take a photo of it, there might be a cheaper solution in repairing it if the conductors are in good shape and only the insulation is abrazed or cut. This given the lamp base itself is in decent condition - not broken and has sufficient tension to lift a lamp with it.
Otherwise if just the lamp base with whip or tails depending upon style, that's standardized parts and you don't need to shop from any one brand. Look to other postings in me recommending the Sylvania brand with aluminum frame around it to protect against breaking. Lots of past info from many sources on PAR can wiring.
I don't think that will comply with your code. Whilst the heat shrink will cover the exposed cable, it is not rated as a replacement for the insulation that has flaked off. Besides, if the cable is damaged in one part, it is most likely that it will fail in other parts as well. Thus replacing the wire will be your best bet.
As Ship suggested, post some pictures so we can see exactly what it is that you are referring to and so that others on this site will also be able to benefit from this topic.
High temperature heat shrink is not flexible. It's only servicable use in in covering high temperature butt splices as needed when attaching the lamp base to a cable whip as opposed to using the leads of the lamp base in a fiberglass sleeve in going to the plug.
Problem it would seem both Mayhem and I have is that we don't really understand by your words which conductor it is that is exposed, which of the above two systems is in use, or to what extent the damage is to which part in recommending a solution of replacement. In most all conditions, replacement will be necessary unless the ground in which case it's possible you can possibly fiberglass electrical tape it due to it being a ground and not mattering if it touches the frame as it does already.
On the other hand, given you had enough slack that some of the conductors got caught in the cap, it might be possible to high temperature butt splice the thing and either cover the splice in either the high temperature tape or use the high temperature heat shrink. This is possible dependant upon the general condition of the cable in question that I would want to look at.
Short of this, I recommend you either replace all infected components, or take it to a qualified service person to have a look at it. Just can't confirm such a thing anyway as what to do.
You only need to replace the socket and wiring, removing ONLY that from the can - leave the groundwire connected to the can and connector. It's common to see a plastic strain relief - know as a keyless strain relief and commonly sold under the Heyco name. You can get this off with a pair of ViceGrips or pliers (or an expensive Heyco tool - $50), just be nice cause you want to re-use it. If the fiberglass outer jacket for the instrument leads is frayed, PA can sell you individual lengths, pre-cut as well. Heyco bushings also if needed.