A google search provided a plethora of instances in which the term was used, but isn't it kind of repetitious? Can you be indignant with something besides rage?
I'd say "he was beside himself with rage" or he was "apoplectic with rage". Indignant and rage, in my opinion don't collocate. "Indignant" implies a quieter or more restrained anger/ resentfulness, while "rage" evokes images of hysterical outbursts. But that's just me.
I do not think I would say 'I'm indignant with rage' for the reason, as someone has explained, that 'indignation' involves a less impassioned state than 'rage'. It would be like saying ' I am pleasantly contented with ecstasy!'
It would make more sense to say 'I am enraged with indignation'.
Sometimes people will say they are 'incandescent with rage' which seems better, since 'incandescence' does involve heating something until it is white-hot, so would seem an appropriate intensifier to 'rage'.
Indignation and rage are concepts that are different but overlapping. Indignation usually implies that an injustice has been done, and it doesn't specify a level of emotion. Rage describes an intense emotion. An angry dog could be described as experiencing rage, but its rage is unlikely to be the result of indignation, since dogs don't really understand justice. "X with Y" usually implies that Y caused X, but indignation would cause rage, not the other way around. "Enraged with indignation" would imply that the person's sense of injustice had been offended, and because of that the person was experiencing a strong emotion.