I see.
Although...
Simultaneously appeared in all eras after 12000 BC.
I don't think that's how the Omen appeared in the other eras.
True. It makes more sense for the Omen to have made a single trip in time.
Though it creates somewhat of stalemate between the ideas that
(idea 1)it was created/teleported to the sky in 12000BC
(idea 2)it was created somewhere in the future and time travelled to 12000BC
because... (idea 1) if we destroy the Omen in 1000AD or 600AD we can fight it again in 12000BC because, as we know, killing a future version of something won't destroy a past version of it.
and, (idea 2) destroying the Omen in 1000AD or 600AD, even if the Omen is in a PD, wouldn't destroy the Omen in 12000BC because of TTI.
So I guess my idea now is something that I hope you guys will find reasonable: (though I'm not claiming anything)
Omen is created/made fully functional in ~2300AD to absorb more energy from Lavos right above Death Peak.
Omen warps back to 12000BC from Time X > 2300AD to prolong how much energy it can absorb from Lavos.
Omen has TTI for its appearance in 12,000 BC.
Omen is destroyed on the Day of Lavos 1999AD.
Omen Crono and Co. see in 2300AD is the Omen pre-time travel.
Speculation, of course. But I don't think its ridiculous. I think it makes more sense than the entire undersea palace being severely reconstructed, having a whole bunch of new technology, and very strong mutants seemingly out of nowhere - all in a relatively short amount of time.