I was always under the impression that the Declaration of Pilnitz was meant more as something to stop the emigres from moaning and the Prussians were surprised when the French government took it so seriously.
Not really. It's why the Russia finally ended its war with Ottomans, that the king of Sweden declared he would take the command of the army.
The Prussians were actually more cautious, yes, and wanted "only" to threaten enough the french to make them cancel many reforms. But they didn't made it only for the émigrés. They tried to both maintain a balance between the ones that wanted to go to war immediately and the ones that wanted to use that to make their power grow in Europe. It happened that, by playing the "middle" and the "entente", they were less busy with French Revolution than the others.
But they were still really worry about it, and the peasant revolt that happened one year before in Saxony was an actual issue.
It's not really an ultimatum, but it's already a declaration of principle against Revolution, its principles and about the capacity of European monarchies to unite themselves against them.
From France, it looked like "Stop it now, give his former power to the king and to the nobles, or we could be less kind".