Prof. John Wall in "Practical Color Photography" (1922 available online), on pp. 77-79 discusses the process and identifies the dyes used in the Pinatype dye transfer:

RED- natural carmine; lanafuchsin BB or SL;

BLUE- indulin blue for blue;

YELLOW-acid yellow, mikado yellow, or quinoline yellow.

Carmine is of course a well known colorant obtained from insects, C.I. 75470, aka Natural Red 4. (See first attached file thumbnail and also It is likely that this is the dye used in Pinatype because the Pinatype dye and carmine both had to be mixed with ammonia before it was mixed with water and used. It is pink in color and would serve for the magenta.

Lanafuchsin BB has Colour Index number C.I. 16630 with this chemical description: 3-[(5-Acetylamino-2-methylphenyl)azo]-4,5-dihydroxynaphthalene-2,7-disulfonic acid disodium salt. There are illustrations of similar chemical structures, see Acid Violet 6, C.I. 16600.

Indulin blue, aka Acid blue 20, c.i. no. 50405. The structure is the second attached graphic.

Quinoline yellow, aka Food Yellow 13, D&C Yellow No. 10, Acid yellow 3, Quinidine Yellow KT, Japan Yellow 203, Lemon Yellow ZN 3, C.I. 47005
IUPAC name[hide]
Sodium 2-(1,3-dioxoindan-2-yl)quinolinedisulfonate

The structure is pictured in the third graphic.

I imagine that the first set of Capstaff Kodachrome dyes were similar in character.