Lutron dimmer w/companion in existing Carter 3-way
i never heard of a carter 3-way. so i looked it up.  I am very sorry.
the short answer: no, rewire the whole thing. carter 3ways where banned in 1923. and are very dangerous.
however, to identify the  wires at each box
with a multimeter 2 wires should read 120v the third is the light to be controlled.
and the hot and neutral go on the &quot;Traveler&quot; screws. and the light(load) goes on the common screw. (this is for the horribly unsafe carter 3 way)
what's happening is you are swapping the hot and neutral. so you have to think. can a dimmer work on the neutral side of a load?  no, no it can't. they only work on the hot side of the load (light).
also. don't change light bulbs without turning off the circuit at the panel.