If I were to hazard a guess, I'd say there are several reasons why you wouldn't want to do this.
I'll reference A Revolutionary Lunar Space Transportation System Architecture Using Extraterrestrial LOX-Augmented NTR Propulsion (describing the LANTR) a few times below. Here's the important bit of figure 5 from that paper:
Firstly, as you suggest, nozzle temperature is almost certainly an issue. It is suggested that the exhaust temperature exceeds 3500K with the cool subsonic oxygen injection, which is already a fairly punishing environment for any material. Some effort was made to keep the exhaust gas temperature under 3600K in the LANTR design, though they don't say precisely why, but I'll bet that as things get much hotter than that your nozzle or your oxidiser injectors will just start burning up, regenerative cooling or no.
Secondly, the post-throat part of the LANTR is already operating as a scramjet, spraying subsonic oxygen into a supersonic hydrogen flow. There are a whole bunch of technological challenges involved in that, ensuring that the fuel and oxidiser mix and burn whilst they're still within the nozzle. What you're proposing involves injecting a supersonic oxidiser stream into a supersonic fuel stream and hoping it all burns nicely before leaving the nozzle... something that sounds a substantially less simple than making a scramjet which we still seem to be having problems doing. Possibly you were thinking of superheated subsonic oxygen injection which seems less implausible from a combustion point of view, but injectors which could do the job sound like a serious engineering challenge.
Finally, you're faced with the problem of handling a 2-3000K high pressure oxygen stream, and no materials engineer is going to relish that. At that temperature, a small amount of the gas will have disassociated into monoatomic oxygen, making it even more reactive than usual. You'll need a different reactor core to cope with the harsh oxidising environment... the hydrogen reactor core won't do at all, it being optimised for a reducing environment, so you need to design two different complete working nuclear rockets to make just one engine. You can't simply hook up the oxygen reactor to the nozzle of the hydrogen reactor... you'll need some sort of plumbing which needs to cope with even more heat whilst carrying very high pressure superheated oxygen.You can't vent both preheated fuels into a combustion chamber and then blow the result out of a nozzle because the chamber temperature will be too high and you won't be able to run your engine efficiently in hydrogen only mode.
That gives you three nightmarishly difficult engineering problems to overcome, for a what isn't obviously a major improvement over a LANTR. If you can achieve three impossible things before breakfast, why not finish up with making a practical liquid or gas core NTR instead?
Additionally there are other performance issues which aren't quite at the level of complexity of those above. You're having to pack in two nuclear reactors now, and if you're not running both of them at once you've got a big dead weight to haul around until you finally decide to use it, which reduces your thrust-to-weight in non-augmented mode. You can't use both reactors as hydrogen-reaction-mass rockets, because of the impracticality of having fuel elements that can work at 3000K for both an oxidising or a reducing propellant depending on your mood.
If you're not running both reactors, you have to switch your oxygen rocket on when you need it, and off again when you're done. Your NTR will have a limited number of cycles, and you'll have to cool it down (by venting propellant through it) when you're done, which costs you reaction mass. This contrasts with a LANTR which can smoothly shift between normal and augmented mode of operation without having to cycle the rocket at all.