You don't have to do anything in particular. The idea that challenging deep assumptions would make you insane is just another assumption. If you believe it, then you can pretend it's true and act as if it was true. But only if you believe it.
If reactions happen, observe them, ask yourself what they are really about. There's no "in-between", there's just the idea of a self that is going through a process in which beliefs are being challenged. This idea of self, of a process are also ideas.
Remember when Ralston talked about how kids practice "make-believe" and become really good at it? In that same book. If you want to avoid actually challenging the things you hold as true, you are fully capable of pretending anything to avoid it. That includes making a story in which there's a self that struggles to "stay grounded" or is scared to go insane.
Don't take it from me though. You can use the contemplation practice Ralston presents in this book and simply ask yourself what this issue you are describing is really about.