Most films lose 2 to 4 stops when exposed through the antihalation base; many antihalation layers are also strongly colored (Foma, Lucky, and J&C Pro 100, for instance, have a dark blue base color before processing). And of course exposing through the base adds an additional complication when using 120, because you'd have to untape the film from the backing, print it, and tape it back the other way around. And then you'd have to worry about putting an impermeable layer over the antihalation (which is commonly coated on the base side, in the anti-curl layer, for films that have this, though some films have it on the emulsion side in a layer beneath the sensitized layer).
However, in addition to Kodak KIE, Maco IR820c Aura (another infrared film) has no antihalation, and can be had in sheet format as well as roll films. It's much slower than HIE in visible light, and a bit slower in IR. There may also be copy films that have no antihalation and are panchromatic, though copy films are generally quite slow (to allow comfortable exposure length under an enlarger or in a contact printer). I had envisioned this in large format; the bigger the negative, the less fine the filter matrix need be to give acceptable image quality, and exposing through the base is at least possible. And obviously the faster the film, the more speed loss can be tolerated due to the filter matrix.
I see potential problems with mordanting the dyes; if done with a bath after printing the filter matrix, but before exposure, the mordant bath is likely to (at a minimum) remove sensitizing dyes from the film, further reducing speed. In addition, I'd be concerned that a mordant might tend to harden the gelatin, which would affect process times. If the mordant is part of the ink, the same problem occurs with potential hardening, though at least one needn't worry about loss of sensitizers.
Most ink jets use either piezoelectric actuators (HP? Lexmark?) or strong, rapid local heating with associated steam bubbles (Canon, Epson, probably others) to fire the ink. The latter is strongly dependent on the ink having a boiling point very close to that of the original formula, but a piezoelectric head could spray (for instance) organic solvents (anything from acetone to xylene, in principle) carrying water-inert dyes that could penetrate the gelatin without washing out once dry (dyes from Magic Markers or similar permanent fiber markers would be a good first hack). Surface tension might be an issue (both in droplet formation and spreading after impact), but can be tailored to some extent in formulating the dye ink.
As I recall, the potato starch in the original autochromes was protected from developing chemicals by the bitumen matrix, so those dyes didn't need to be water inert.