There probably is no one best approach, but before dismissing boycotts as unlikely to be effective, remember the global boycott my generation organized to force the South African government to end Apartheid. That boycott was much more than a few tourists deciding to shop elsewhere. It was an international shunning of almost everything South African -- that brought the South African economy to its knees. It subjected ordinary South Africans to practical travel and education restrictions as well, as corporations and universities all over the world refused to do business with South Africa or South African people.
I don't think anyone disagrees that this powerful boycott that turned South Africans into international pariahs helped force the collapse of Apartheid -- and far faster than would otherwise have happened.
Something else to consider here is the idea that homophobic regimes necessarily track with poverty, and that wealth changes things for the better. Russia and Poland are two powerful counter examples. Both nations are becoming more hostile to LGBTQ people, enacting harsher laws, and seeing increasing rates of anti-LGBTQ street violence even as standards of living are surging and individual wealth is increasing.
Increasing prosperity has tracked with upsurges of homophobia in many other nations, including Nigeria, an extremist anti-LGBTQ state I wrote about in the article you linked here. People in Nigeria do not typically struggle to earn enough money to eat or live. And much of the sentiment toward enacting harsher laws originates in relatively prosperous urban centers, not the poorer countryside and villages.
Cameroon is closer to the picture you paint - of a developing nation whose people tend to be very poor and even to struggle for subsistence sometimes. But even in Cameroon, anti-LGBTQ sentiment is harshest and most centered in wealthier populations - like my friend Joseph's family. He started a business in the U.S. because his family are rich and powerful in Cameroon - able to provide him with seed capital for his import/export firm.
Right now, the EU is ramping up toward enacting harsh economic sanctions against its member state Poland in order to work toward forcing Poland to stop persecuting LGBTQ people. I wholeheartedly endorse those potential sanctions, hoping they will be as strict and punitive as possible. I hope (though I doubt) they will be as punitive as the ones that gutted South Africa's economy. Because it's not OK to hurt marginalized minorities like innocent LGBTQ people. Force and power are effective when used to protect minorities. Force and power must be used when they can be appropriately harnessed to create positive moral change.
