TL;DR: The bureaucracy operated from St. Louis and didn't really get out to Dubuque, relying instead on the expert opinion of a local trader. Dubuque was the active party in pursuing the grant as he wanted to protect his de facto property legally. Also, while 300 miles might seem far, Dubuque went to St. Louis twice a year to sell his wares and so was in relatively close contact with the authorities.
Background:
Julien Dubuque was born in 1762 near Quebec City. A roaming adventurer, he emigrated to Spanish Lousiana in 1774, fur trapping, prospecting and travelling around the territory before ending up in the village of Prairie du Chien, Wisconsin in 1785. This little town, the second-oldest in Wisconsin, was located on the eastern bank of the Mississippi and therefore technically just inside the borders of the United States since the 1783 Treaty of Paris, but Dubuque's operations seem to have concentrated mostly across the river in Spanish Louisiana.
Over the next years, Dubuque befriended members of the local Mesquaki tribe (the 'Fox Indians'), who called him 'Little Night' ('la Petite Nuit'). The native tribe led by 'Kettle Chief', from 'Little Fox village', just south of the current city of Dubuque, operated a small lead mine during these years near Catfish Creek, discovered earlier by a Mesquaki who is described as 'the wife of Peosta'. The mine piqued Dubuque's interest, and he and his followers established themselves at Little Fox village, studying the mining in progress there.
Impressed by what he discovered, Dubuque spent the next years buttering up to the natives, doing everything he could to flatter and intimidate them into selling him the mine and the surrounding area. Apart from presenting the tribe with gifts, he is 'said to have resorted to tricks of necromancy' and claimed immunity from snake-bites through covert use of antidotes. Most famously, when the Mesquaki still refused to give him the mining rights, he is said to have threatened to set fire to Catfish Creek, and then followed through on his threat by having one of his associates dump a barrel of oil upstream, Dubuque setting fire to the oil once it reached the village.
On 22 September 1788, all these efforts reached their conclusion. Dubuque hosted a council of local Mesquaki chiefs and braves in Prairie du Chien, who agreed to 'permit Mr. Julien Dubuque, called by them ‘Little Night’, to work at the mine as long as he shall please, [..] moreover, that they sell and abandon to him all the coast and the contents of the mine [...] so that no white man or Indian shall make any pretension to it without the consent of Mr. Julien Dubuque.'
Satisfied with this treaty, and allowed by the natives to work on the mine in peace, Dubuque set to work extracting lead from the mine. He enlisted the help of Little Fox village and other tribes in the area: squaws and old men (the braves were above working) helped him build a long house, horse mill and smelting furnace, and did most of the actual mining. The lead mine was a major success, and soon Dubuque was extracting twenty to forty thousand pounds of lead per year, floating it down the river to St. Louis twice a year where it was sold.
The semiannual arrival of Dubuque, 'the first white man in Iowa', and his little lead armada was quite an event in St. Louis. Dubuque, described as 'well-built, capable of great endurance, and remarkably courteous and polite, with all the suavity and grace of the typical Frenchman' mixed in well with the local upper-class socialites, who on occasion gave balls in honour of his arrival, and he soon became something of a local celebrity.
Obtaining the grant:
It was likely at this point (1796, around eight years after opening the mine) that Dubuque started to doubt the legality of his Indian treaty, and wished to protect his newfound wealth by having confirmed. Most likely he was looking to the future: he would sell a portion of his grant to a third party some years later, and the wording of the treaty with the natives was such that it was not quite clear whether he had bought the land or just the mining rights. And so after this long introduction we come to the crux of your question: how did Dubuque receive his grant?
The answer is: two letters and consultant Andrew Todd.
The first letter was a petition written by Dubuque and presented (possibly in person) to the Governor of Spanish Lousiana, Baron Francisco de Carandolet, in St. Louis on 22 October 1796. Dubuque was an excellent groveller: in the petition, '[this] very humble petitioner prays your excellency to have the goodness to assure him the quiet enjoyment of the mines and lands' which he claims he 'has bought' from the natives, 'and be pleased to accept the pure simplicity of my heart in default of my eloquence. I pray heaven, with all my power, that it preserve you, and that it load you with all its benefits; and I am, and shall be all my life, your excellency’s very humble, and very obedient, and very submissive servant'. As if this sycophantic petition wasn't enough, Dubuque also named his mines the 'Mines of Spain' in a further attempt to ingratiate himself.
Baron de Carandolet of course had never been to Dubuque's mines, so he referred the petition to Andrew Todd, a trader who holds a monopoly on trade with the natives in the area around Catfish Creek, 'with direction to Todd to inform him, the Governor, as to the nature and propriety of Dubuque's demand'. Todd responds favorably to the governor, stating that he sees no reason not to grant the land, on the condition that Dubuque does not interfere with his monopoly by trading with the natives. Satisfied with the answer, Baron de Carandolet sends the second letter on 10 November 1796, accepting Dubuque's ownership of the land on the condition that he does not interfere with Todd's trading monopoly.
Epilogue:
The story does not end there though! After the Lousiana Purchase, governor of the new Indiana territory (and later president) William Henry Harrison arrived in St. Louis, and wasted no time in arranging the controversial Treaty of St. Louis (3 November 1804), where the United States seized a big chunk of Mesquaki and Sauk land in exchange for a pittance. The Sauk in particular were furious about this treaty, and fought the Americans under their leader Black Hawk during the War of 1812 and the Black Hawk War of 1832. Back in 1804, Dubuque lobbied Harrison to add an additional article to the treaty, stating specifically that any arrangements made during Spanish rule with the consent of the natives (including Dubuque's claim) were to be upheld. Two years later in 1806, Harrison even explicitly confirmed that this clause was added to protect Dubuque's land.
Why would Harrison do that? Well, no proper survey was done on Dubuque's claims, which as it turned out included a large native village which it had probably not been the natives' intention to sell. During my research I found a number of lawsuits made against those who had purchased land from Dubuque (who died in 1808 or 1809), so most likely Dubuque was already fighting to protect his claim during his lifetime, and his heirs and buyers were fighting it up to the 1840s and 1850s! One of the buyers of Dubuque's land, a Henry Chouteau, even petitioned the 27th US Congress, leading to a 37-page report that is the source for much of this.
So, how did bureaucracy function so far from St. Louis? The answer is, by letters and expert opinions. Did it function well? Not particularly. The survey system was not properly rolled out so far from the capital, and it was not clear who even had the authority to grant Dubuque his land (the natives or the Spanish).
Sources:
Brigham, J. Iowa: Its History and Its Foremost Citizens 21-29. Chicago: Clarke Publishing, 1918. Retrieved from Iowa History Project.
S.350 – 28th Congress, 1st Session (1844). Report: Petition of H. Chouteau (3 June 1844). Retrieved from University of Oklahoma.
Kappler, J. (compiler). "Treaty with the Saux and Foxes, 1804." In Indian Affairs: Laws and Treaties 74-77. Washington: Gov. Printing Office, 1904. Retrieved from University of Hong Kong.
Chouteau v. Molony, 57 U.S. 203 (1853). Retrieved from casetext.com