One of the first things the Confederacy did after the war started was to invade Federal territory in the Southwest to secure it as slave territory. I assume you must have heard of Mesilla? Confederate forces took it 103 days after Fort Sumter fell, with an expedition that set out less than two weeks after Sumter.
Moreover, the Confederacy in its early days - back when most Confederates believed they had a chance of winning the war and dictating terms to the Union - would likely have demanded the return of fugitive property and contrabands as a condition of a peaceable settlement.
I can't imagine than an independent Confederacy would have signed any treaty that did not compel the Union to return escaped slaves - a grievance that was repeated again and again in secession conventions.
You and I have discussed this before. I see you have now moved the date of the expedition to after Fort Sumter rather than before Fort Sumter as you've posted in the past. That is progress of a sort, but you are still off in your time table. On May 27, 1861, Colonel Earl Van Dorn said the following [Source]:
Within a short distance of El Paso or Fort Bliss there are several hundred United States troops. I have, therefore, ordered four companies there. There are five or six pieces of artillery at Forts Davis, Quitman, and Bliss, which I have ordered to Fort Bliss.
The book "The Confederate Invasion of New Mexico and Arizona" by Robert Lee Kerby provides some timeline information on pages 33-34:
... in the first week of July, Balyor's battalion of the 2d Texas Companies A, B, D, and E) and Captain Trevanion T. Teel's battery B, 1st Texas Artillery, occupied El Paso and took quarters in Fort Bliss.
As you may know, El Paso and Fort Bliss are in Texas, not New Mexico. I gather Baylor later sent small detachments into New Mexico in July to gather information about Fort Fillmore which was very near Mesilla, New Mexico.
The people of Mesilla were already flying Confederate flags. Mesilla was the place where two conventions had earlier (Feb 3 and March 16, 1861) declared the secession of the Arizona Territory which was then roughly the southern half of present day Arizona and New Mexico. They wished to be a territory first of Texas (Feb 3) and then of the Confederacy (March 16, see Link).
Baylor marched on Mesilla July 23-24, 1861. The federal commander of Fort Fillmore decided he had no chance and abandoned Fort Fillmore on July 26. He and his troops were captured by Baylor on July 27.