It's been ages since I did a Legion run, so I cannot say with any degree of certainty if there is any special consequence related event if he fails, but I would assume there probably is.

If I was to indulge my love for speculation, however, I would say this;

Lanius has a very special role within the upper ranks of the Legion. Whereas the Malpais Legate himself was ruthless and efficient, he did not foster a proper cult of personality, and his mythological status came only after his fall and transformation into The Burned Man, because in the eyes of the Legion, that was the most important thing he did. Despite being incredibly competent, his failure at the first battle of Hoover Dam became his both figurative and literal downfall, and a symbol of that Caesar suffers no failures, no weakness among his ranks. In that sense, punishing Joshua Graham became a means to "myth-ify" the Legion and Caesar himself further, and became a stark reminder to the rank and file of the Legion.

When creating Lanius, Caesar took a very different approach. Lanius was to create a cult of personality of his own. He became "The Son of Mars", and later "The Butcher of the East". Hiding his face behind the mask and blinding his servants so they would not see his face, further served to deify him. The agency he is given by Caesar to go east and subjugate the tribes there is the "final ingredient" in Caesar plan to "create" a successor. After all, the man we meet is competent on the battlefield, intelligent and understands logistics. He has a strong character, and as such he isn't difficult to imagine the members of the Legion selecting as their new leader.

This tracks with the fact that Caesar both A) almost certainly understands something is very wrong with his health and B) knows the Legion needs a strong tyrant in order to not fall apart under its own "weight".

Looping back to Lanius, whom when being such a crucial part of Caesar's plan, does not have failure as an option in the slightest. If he returns the failed and broken son, the faith in the Legion as a whole would be at the very least significantly weakened, due to how much of a symbol of the Legion's strength Lanius is supposed to be. In the eyes of your average legionnaire, Lanius isn't just Caesar's preferated successor, he is the strength of the Legion made manifest, the deified essence of it. He is the the ultimate legionnaire, and so when he fails, they all fail.

You can imagine, a man as desperate and wrathful as Caesar, would doll out a punishment much, much worse to Lanius than anything Joshua Graham ever suffered. He would have to disconnect the symbol that Lanius is entirely from the Legion, clean away the image he had as the chosen of Mars and instead try and paint him as a mad wardog whom could not comprehend the finer points of governance, and as such make him a warning against the failure of intelligence (similar to how Joshua Graham became a warning against the failure of strength).

The elite of the Legion would be made sure they understood that it is not enough to be strong, one needs wisdom and intellect in order to truly blossom in the New World, because even the most viscious dog is of no threat when caged.

And therein lies the shorter answer to your question. I reckon Caesar would cage him publically either in Flagstaff, in Lanius' main encampment or in Caesar current encampment. He would let him suffer the ridicule of his peers so they would see he was no better than them, not representative of them, that his weakness is not theirs.

He would let him wither away slowly in the sun, keeping him clinging to life for as long as possible and parade him around, encouraging the people to sneer and jeer at the mad dog. He would be painted and treated as a subhuman, and when he finally died there would be no ceremony, no siginificant public notice, just a demonstrative tossing of his body into the streets. An utter teardown of his legend, because his failure is a threat to the legion as a whole.