The Roman Empire became a Christian empire before it collapsed. The collapse took hundreds of years. Whatever micro analysis we could do of a decline in morality is not going to reveal as a cause some kind of pagan anything goes homosexuality.
The causes of the empire’s decline and fall are many but certainly it succumbed to foreign invasion. And of course, The Eastern Roman Empire survived an additional 700+ years.
One of the biggest reasons Rome declined and fell was because it never solved the “transition to a new emperor problem”. It essentially was a military dictatorship to the bitter end. It tried to maintain the ‘fictious’ notion that it was still in some way a republic. This made the monarchial principle of the state being the personal property of the ruler where it stayed in family with no outside input difficult. They tried but it never worked well. That type of transition was what barbarians do. The self image that Romans are Romans not uncivlized barbarians carried on to the end. The appearance that the emperor became emperor through the approval of the military, the Senate and the people of Rome had to be at least visually maintained. The transition in reality was always something cobbled together. Hence the many civil wars over who had control. These civil wars sapped money, manpower and economic stability out of the ‘system’.