Learn English – Except only + clause

From each of your proposed sentences, I would assume that you have lots of time on Sunday, and very little time on any other day. If you want to convey that you have no time at all on any other day, you should say,

"Sunday is the only day when I will have time available to do the homework."

I'd say they're questionable at a more complex level than pure syntax. Obviously, '[sentence A] except that [sentence B]', with punctuation adjusted as required, cannot be claimed to be non-syntactical – there is an example from AHD below. This usage of except that as a conjunction means

but for the fact that [= 'only' when used in this sense][a comma may be necessary] [Collins]

I would buy the suit, except that it costs too much. [AHD]

If the Collins definition is exhaustive, 'He is nice except that he is a little shy.' is wrong because 'He is nice were it not for the fact that he is a little shy.' is wrong. I'd prefer 'only' here, in its 'but there is a minor negative:' sense

He is nice, only he is a little shy.

And 'only' can't be fronted here. 'Although' could be.