If the Emancipation Proclamation applied to a “separate nation,” then there would be no constitutional grounds to oppose it.
Exactly. That is why it was necessary to post both the Union position and the Confederate position. Lincoln never accepted the “states in rebellion” as a separate nation.
In practice, every time Union troops gained control of an area in the Confederacy, any slaves that made it to Union lines were free. That was in direct opposition to federal law, the Fugitive Slave Clause, Article IV, Section 2 of the Constitution and the Fugitive Slave Act which required return of runaway slaves to their masters.