I cast a quick glance at your sentences and I thought the task was about to prove in #1 that run and up belong together (verb plus particle), while in #2 that we have a verb plus a transitive preposition.
Your first replacement test does not work, in my humble opinion. If you substitute the proform for the string of words after 'run', the meaning will not be the same anymore. It is important to note with regard to your tests that they work in one way. This means if your test confirms your initial assumption, it means you were right. If your test does not corroborate your theory, you are not necessarily wrong.
The proform replacement test should look like this:
The businessman run up a huge bill.
I suspect that the sentence should be carved up into constituents as the brackets show:
[The businessman] [run up] [a huge bill].
The businessman run it up.
RNR. Do you know what that is? Right node raising. This is a test for NP constituency, and a right movement. It looks like this:
The businessman run up _, and her wife agreed to foot _, the bill.
What did you run up?
A huge bill!
It is a huge bill that he run up.
HE [run up] a huge bill.
He [incurred] a huge bill.
My assumption vis-a-vis #2 (is the mother of all...) is this:
They run [down the alleyway].
They run up the alleyway.
They run there.
It is up the hill that they run.
Up the hill they run.
Up a bill the businessman run.
Peter run up the hill, and Marta (run) down the lane. -- run and down are not married.
down the alleyway --> PP
Student or Learner