More to the point, did Louallier ever get his writ of habeas corpus?
I don't know the answer to your first question. For your second, it does not appear that Louallier got his habeas corpus. Apparently he was held in jail by Jackson even after being found innocent by a Court Marshal and was only freed after news of the end of the war reached New Orleans. I don't think Louallier was held in jail very long since it appears that the offense for which he was held occurred some two months after the Battle of New Orleans.
From an 1841 New Orleans City Directory:
The war with Great Britain followed immediately afterwards. With all its incidents, ending in the GLORIOUS 8th of JANUARY, our readers are familiar.
About two months after the British had retired, Louallier, a member of the Legislature, made a publication against an order of General Jackson, removing to one hundred and twenty miles beyond the city, a number of Frenchmen, who, after joining bravely in the defense of the country, had obtained certain certificates of their national character, from the French Consul, for the purpose of leaving the army, which was still kept under arms, and returning to their families.
This was immediately followed by a writ of habeas corpus, issued by Hall, the Judge of the United States court, for the release of Louallier. Whereupon, Judge Hall, in turn, was also imprisoned. Louallier was acquitted by a Court Martial; but neither he nor the Judge were released until official news of the peace had arrived.
Hall had no sooner gained his authority as Judge, than he punished Jackson for a contempt of Court, by fining him one thousand dollars, which was paid by the people of the City.
Years later, the US Congress paid Jackson's fine with interest. I wonder whether the people of New Orleans ever got their money back.