Interesting. Did the Fascist Grand Council have that power? It would mean Mussolini was less than an absolute dictator. I don't think that was the case in 1939. I think Mussolini was waiting to see how the war evolved. When it was clear France was defeated, he invaded and took a tiny piece of the French Riviera. In any case there was nothing corresponding to the FGC in Germany.Fair enough.
I do think that it's possible that Hindenburg would have intervened in such a scenario just like the Italian King and Mussolini's minions on the Fascist Grand Council could have removed him from power had he declared war on Britain and France in September 1939.
The Weimar Constitution gave the President the power approve and dismiss a government. However the Enabling Act essentially took that power away. There was only one party and one leader. If he did dare try to interfere , how would he do it? He had no levers to pull. If Hitler did find the President troublesome he could simply have a nurse apply a pillow to his face and solemnly announce the revered President died in his sleep.