These days, such a major arms purchase would need end user certificates, all sorts of permits from a web of law enforcement agencies and the DoD, if not approval from Congress. Oh and I'd probably need to be a nation state (I think there's a seminar to become just that). But what kind of bureaucracy would I have had to deal with at the end of the 19th Century?
I can't answer your question but I would like to point out that by 1896 the Gatling gun was obsolete, as Maxim had already developed the modern machine gun by 1884. The same gun that was used to devastating effect by both side in WWI.
I'm more of a Cuban history enthusiast, and this seems more aligned as an American gun control history question, but I'll share what I've picked up based on reading two sources:
There was really a" laissez-faire" attitude from the US Government regarding arms sales from weapons manufacturers up until the end of World War I, when the carnage of the conflict began turning public sentiment against these newly minted "Merchants of Death."
In fact, due to the development of an imperialist foreign policy at the end of the 1800's, a weapons deal from American gun manufacturer(s) to Cuba would no doubt have been encouraged. America already had a more than subtle interest in an independent Cuba (ripe for American corporate takeover). One lawmaker during this time period was quoted as saying "Fate has written our policy. The trade of the world must and shall be ours."
So from an economic standpoint, it would seem that a Cuban businessman with enough capital could indeed purchase heavy weaponry easily from a manufacturer, providing it would probably have to be assembled and moved from up North down to somewhere like Florida, where the weapons would be loaded into ships. It would cost a very large sum of money to accomplish this.
On a side note, it would seem not too many native Cubans would have had the type of capital needed to support this level of arms acquisition. Most wealthy Cubans lost their holdings upon the abolishment of slavery in 1886, which decimated the sugar trade on the island and saw much of the wealthy class get knocked down to middle class status. Now an American businessman who had spent significant income on building up holdings in the sugar or tobacco industry in that vacuum though...perhaps with ties to the mob...(read Havana Nocturne!).
Looking at gun control laws of the time period also points to a lack of bureaucratic hoops for an aspiring "independent Cuba" buyer. The first major national gun control legislation didn't occur until 1934 under the "National Firearms Act of 1934." Several states had laws regarding gun ownership or carrying regulations (mostly to curb African Americans and immigrants from owning or using guns) but these laws would not have applied to foreign arms sales. There seems to be an utter lack of oversight in this area by the US Government up until the early to mid 20th century!