Ah, okay. Bullet point answers:

• IF Assuming Theism was not true
• AND Assuming no fear of losing Theism if Theism was not true, then
• THEN I would drop Theism as soon as I convinced myself it wasn't true

Other variations on the above format:

• IF Assuming Theism was not true
• AND Assuming there is fear of losing Theism if Theism was not true, then
• THEN I would drop Theism as soon as I convinced myself it wasn't true
• ONLY IF I overcame my fear of losing Theism.

I would expect convincing myself Theism isn't true would be harder than overcoming my fear of losing Theism. This leads into your question:

To what extent does this fear compete with your desire for accurate beliefs?

You are implying a scenario more like the following:

• IF Assuming Theism was not true
• AND Assuming there is fear of losing Theism if Theism was not true, then
• THEN I would convince myself Theism wasn't true
• ONLY IF I overcame my fear of losing Theism.

Which is a subtle but important difference. I like to think that my fear wouldn't cloud my ability to perceive the truth... but I don't actually know how to verify that. Signs seem to point the exact opposite way, in fact.

I suppose one solution would be to lesson my fear in losing Theism, which seems to be the route pjeby suggested in another comment.