In the United Kingdom, experiences in overseas possessions (principally on the Indian subcontinent) not only inspired a wide spectrum of writings (ranging from E.M. Forster's A Passage to India to George Orwell's Burmese Days) but also created a whole Anglo-Indian subculture among the mostly middle-class civil servants and officers who entered the British military/colonial administration.
Likewise, references to Algeria are common in 19th and early 20th century French literature. In Maupassant's Bel-Ami, protagonist Georges Duroy is a retired soldier who has spent significant amount of time in North Africa and later manages to establish himself as a sort of authority on the matter, while the Algerian city of Oran is a favorite setting of Albert Camus, who himself was a French-African.
The American Military Government in the Phillipines and the subsequent Insular Government of the Philippine Islands (and Commonwealth of the Philippines) on the other hand doesn't seem to have spawned significant literature. In addition to running the Philippines, the Bureau of Insular Affairs also managed the islands of Cuba and Puerto Rico; I would think that some bureau officers must have contributed something or other to the american public consciousness, but that doesn't seem to be the case. Why didn't the American colonial community spawn a significant subculture?
One of the most important differences between the examples you cite and the case of Cuba is that we didn't rule over the island until as late as 1898. Puerto Rico either, I might add. Even when we did occupy Cuba, it was only sporadically.
We did have small, commercial communities in important cities in Cuba as early as the 1820s (Cardenas, Havana) but these were limited in scope and more commercial enclaves than true communities.
After Cuban independence, some companies had limited success in essentially copying and pasting the urban model of the American West into the context of Cuba, but such towns didn't tend to succeed, and in any case we're talking about the 20th century while your question is about the 19th. [Here is an article on the subject that I was able to find in English: http://www.jstor.org/stable/2518425?seq=1#page_scan_tab_contents]
Great American literary figures, like Hemingway, did start arriving and writing classics like The Old Man and the Sea, inspired by witnessing old Cuban fisherman, but only when the American presence on the island really expanded beyond utilitarian economic enclaves.
Roland Ely, author of Cuando Reinaba su Majestad el Azucar (When Her Majesty Sugar Reigned) wrote his story of the rise of the Cuban sugar industry using many 19th century American travelogues (even then, people became self-proclaimed experts after just one trip), but no fiction to my knowledge.
I am aware of political poems about Cuba, popular in the 19th century, but these are usually of more use to historians than they are meritorious of literary recognition. You do have exceptions, however, such as Walt Whitman's Liberty Poem for Asia, Africa, Europe, America, Australia, Cuba, and The Archipelagoes of the Sea, but as you'll notice Cuba is perceived as a far away territory he wishes to be free and not a familiar place which he is giving us a poetic tour of using his verse. http://www.whitmanarchive.org/published/LG/1856/poems/21
My point here is that Cuba was, by and large, still a very foreign and distant destination. American presence was limited to small communities who ran commerce or plantations, but not large, settled, communities. The reason being that Cuba did not become an actual colony, and thus Americans weren't exactly welcomed, until the last two years of the 19th century. Once Cuba did become independent, we do begin to see larger communities and a greater familiarity with realities on the Island.
In Cuba, the US wasn't a paradigm either, to be clear. While American political thought was emulated very consciously and served as the source of many subversive ideas, in terms of culture most Cubans looked instead to France. This is true until, honestly, the 20th century. Touching on this point is a fairly well done trilogy called Historia de la Literatura Cubana, published by Letras Cubanas, in Cuba, between the early to mid 2000s.