No person held to service or labor in one state...
For the purposes of our example, Kentucky.
...under the laws thereof...
Said laws allowing slavery.
....escaping into another...
Escaping, not being brought. Escaping. Fleeing. Illegally beating feet and leaving Kentucky.
...shall, in consequence of any law or regulation therein...
For the purposes of our example, an Indiana law prohibiting the ownership of slaves by residents of Indiana.
...be discharged from such service or labor, but shall be delivered up on claim of the party to whom such service or labor may be due.
In our example, the owner in Kentucky. Where slavery is legal. As opposed to Indiana. Where slavery is not. Therefore the captured runaway is returned to its owner in Kentucky, not to its owner now living in Indiana, because slave ownership is illegal there.
As further proof of how fallacious your argument is, look at the Confederate Constitution. It found the need to modify their own article IV, section 2, clause 1 to read: "The citizens of each State shall be entitled to all the privileges and immunities of citizens in the several States; and shall have the right of transit and sojourn in any State of this Confederacy, with their slaves and other property; and the right of property in said slaves shall not be thereby impaired. Likewise they saw the need to modify section 2, clause 3. In addition specifically identifying slaves, where the U.S. Constitution had merely said, "escaping into another" the Confederate leaders felt the need to clarify it by saying "escaping or lawfully carried into another." Why did they see the need to make those changes if a clear reading of the Constitution meant that no state could outlaw slavery completely within its borders?
You are forcing onto the verbiage more specificity than the actual words convey. You are liberally interpreting it with additional requirements not in evidence from a plain reading of the text.
I'm not going to address your foray over into the Confederate Constitution because it does not have the slightest effect on what the US constitution said.