There is no doubt that there are numerous souvenir, commemorative or patriotic pins, badges, etc. in the American design patents, and how to find them is explained at the end of this post. The British registered designs, the same sort of right for the look of a design, are not online for historical material, and do not cover such subjects. They also lack explanatory text which could be searched in this way.
The background to the Spanish-American War is that Cuba, a Spanish colony, was fighting a war for independence from Spain. The American battleship Maine was moored in Havana harbour when, on the 15 February 1898, she blew up with the tragic loss of 266 sailors. Sabotage, screamed many American newspapers (although the evidence is inconclusive), which whipped up patriotic enthusiasm which led to war being declared (by the Spanish) in April 1898. The fighting was mainly in Cuba and the Philippines, and ended swiftly, in August 1898.
What follows is a list of the design patents that I have identified as being relevant, in order of being filed at the US Patent Office. Some of the images are attractive; others are dull.
In fact before the war those wishing to express pro-Cuban sympathies were assisted by Henry Caldwell of Hartwell, OH with his "Badge", D27621, filed 30 January 1897. It shows the Cuban colours and is meant to "indicate sympathy with the cause of Cuba", says the patent.
On the 25 April 1898 Julius Becker of New York City filed for D28787, "Pipe", in the appearance of the Maine.
On the 29 September 1898 Conrad Stein of Bridgeport, CT filed for D29649, "Game board". The text says that the Spanish and American flags are above the forts.
On the 30 August 1899 Warren Greveling of New York City filed for D32001, "Lamp body". It shows Dewey.
Finally, on the 8 March 1900 Anson Bacon of Halliwell, ME filed for D32728, "Pencil". It shows Dewey.
The slogan "Remember the Maine" only turned up once -- as an example of what could be used in setting up type, in "Printing form". Filed on the 7 July 1898, it was published as utility patent 626649.
I found the designs on Google Patents Advanced by specifying a time period and designs and then searching the accompanying explanatory text for each design by keywords such as Dewey, Cuba, etc. The database seems to contain a full set of design numbers for showing PNG images of the drawing pages, but as the remainder is scanned content it is likely that some material has been missed. Of course, if no keywords were mentioned then relevant Design Patents won't turn up. There is a classification scheme, but this would simply define badges, pins, flags and so on.
A specific design can be found on that database by asking for e.g, D27621.