Do I qualify to travel UK > Cuba via Miami as a non US citizen under the ‘family member’ general licence category?

My travel plans for a 4 week visit to my fiancé in Cuba at the start of October are looking in trouble given that my flight to Holguin is booked with Thomas Cook Airlines https://www.bbc.co.uk/news/business-49761464 The options for finding another flight are limited as most would require a bus journey to my ultimate destination (Santiago de Cuba), but the main bus service operator has suspended services for a month due to the current fuel crisis in Cuba.

I’m aware of restrictions on travelling to Cuba from or via the US as a tourist, as explained in the answer to this question Travel from the US to Cancun with a layover in Cuba as an European citizen

My questions are:

• as a fiancée, would I qualify to travel under the general licence category of ‘family visit’ https://www.united.com/web/en-US/content/travel/destination/international/cuba.aspx? As on my two previous visits I will be switching from my 30 day tourist visa to a family visa when I arrive in Cuba, and I will be staying in my fiancé's family home during my trip. Or do I need to be a ‘close relative’ (ie. married) to qualify?
• if I do qualify, what proof would I need to carry with me and how would I go about obtaining the relevant visa in order to board the American Airlines flight in Miami?

We have been in a relationship for almost eight years and my fiancé has visited me in the UK three times; I can prove I support the family but otherwise there is no official paper to evidence our connection.

I do not have anything to prove my two previous family visas as these are stamped on the reverse of the tourist visa, which is a loose piece of paper that has to be surrendered when you exit Cuba. I do however have many entry/exit passport stamps. I don’t want to book the trip only to be denied boarding by AA in Miami or when I try to check in for the return flight.

Note: I don’t need to declare anything about the reason for my visit when flying from the UK direct. The airline just issues passengers with the tourist visa when we board in the UK. I apply for the family visa in Cuba (I can’t apply from the UK because we’re not married); as the owner of the family house my ‘mother-in-law’ accompanies me to the Extranjería along with my fiancé and then I have to wait 72 hours (staying with him in a b&b) while the authorities do their checks on us both. When I get the family visa I can legally go to stay with my fiancé and his parents in the family home.