I think the best thing to do would simply be to bring the debate constantly back to challenging their base assumptions, not to moral arguments that assume your own world view. Anything that builds on both people having the same world view on the subject matter will fail automatically because that’s just not the case.
In the abortion debate for example, no person who is convinced that an abortion literally kills a living, conscious baby is ever going to accept the “my body my choice” argument or the “you just want to control women” one, because obviously bodily autonomy ends at murder, and they believe that these women willfully commit murder. That’s why that argument won’t go anywhere - no reasonable person would believe that anyone has the right to murder someone else on a whim due to their own “bad decisions”, let alone a baby.
That’s why, to truly convince people, you need to show or demonstrate how their base world view is wrong. This is very hard and nigh-impossible, but there is no other way. Someone who is convinced that trans people groom children to participate in their crossdressing fetish will never accept any arguments that are built on the view that trans people are ordinary people who just want to live their lives - because sexual abuse are crimes with victims.
They’re simply not talking with each other on similar grounds.
This is all not to say that you should tolerate or feel pity for these opinions - you shouldn’t - but if someone is actually set on convincing their political enemies of their side, they need to begin at the very base assumptions because they differ wildly.