I think Sturgeon does understand how divisive this issue is. I also think she doesn’t really care a jot about trans people.
one of the problems for the SNP has always been ‘what difference will it make?’ - why should people vote for independence when it won’t really change their lives? What the SNP is trying to do, with this and other issues, is make Scotland seem like a different country to England, with different laws, rules, morals, systems, etc. Sturgeon believes this is an issue she can exploit to prove that Scotland is different and needs to leave the UK.
Women and trans people are pawns ready to be sacrificed for her ambition to be Scotland’s first Prime Minister (or even better, President).