I can not for the life of me, see why Prussia would give up any of Silesia to the Austrians.
It formed one of the core states of Prussia, and was considerably more valuable in minerals than any of Jutland.
Silesia was also eminently defensible against Austria, its mountainous terrain and massive fortresses were too valuable to just give to Austria; especially as Schleswig Holstein was easily annexed by Prussia so why buy it at a terrible cost to Prussias defence?
In August, soon after the war, France demanded Prussia relinquish Mainz - that or go to war with France. Bismarck replied "so it's war then" and the French backed down. Bismarck was no fool, and chanced his arm with Austria in the sure knowledge that France was not ready to go to war in Germany. He always sought the knowledgeable General Staff's opinions, and they considered the risk worthwhile.
If France could have intervened, it would not change the course of Sadowa, and as already been stated, when they tried to, Prussia was ready willing and able to prosecute a war with France.
Indeed it was said that the French Emperor feigned sickness so that the threat to Prussia could be blamed on "the enthusiasm of his ministers" and was thus able to avoid a war he wasn't ready to prosecute.
One must also bear in mind where the Prussian army was at this point - it was sat in South Germany (Mainz) and Bohemia, therefore its route to France would see it stomping on Bavaria before the French could mobilise support, then march into France holding to both banks on the river Maine, avoiding the more traditional route through the cockpit of Europe and the French fortresses of that region.