I just think the whole thing escalated when it didn’t need to. The relationship sounds like a shambles, OP doesn’t seem to like her husband, and he can tell, so why are they together.
They need to go their separate ways, he needs a partner who can understand his autism can cause some rigidity, and she needs someone who doesn’t care whether she respects them.
The relationship now will never improve. He knows she doesn’t respect him, so he spits out his dummy. She lies to him, so he doesn’t want to entertain her. It’s not a functioning relationship, and maybe it could have been if the OP was more understanding, it wouldn’t then escalate to issues like this.
If you are doing something that you need to lie to your spouse about then you shouldn’t be doing it, simple. The problem is, we don’t know what came first, the OP not caring about her husband, or her husband becoming more awkward in his behaviour…given the way the OP words the thread I think she is more likely to have caused this, but either way they just aren’t a good fit.