This isn't how it works.
No doubt, some will try to spin it that way. But the Fed balance sheet's gains and losses are an accounting artefact. (It will always make money when lowering rates. It will always lose money when raising them. This is close to mathematical fact.) The Fed can't go broke. Its operating metrics are inflation and unemployment.
> the selling would likely push mortgage rates up further, at a time the housing industry is already starting to groan under the pressure of rising rates
This is the real issue. But it's why we have an independent central bank. Nobody likes rate rises. It's why economies with weak institutions wind up hyper inflating. (No, we're not.)