For purposes of CIP, is a "TIN application" acceptable at account opening for a non-U.S. person?

No.
You have to get the passport.
The fact that they have applied for a TIN is just informational to you.

Is a W-8 form (U.S. tax withholding) acceptable as CIP identification for a non-U.S. person?

No, that would be insufficient.
If the individual is a non-U.S. person, one or more of the following: a U.S. taxpayer identification number; a passport number and country of issuance (unexpired); an alien identification card number; number and country of issuance of any other (unexpired) government-issued document evidencing nationality or residence and bearing a photograph or similar safeguard.

That was my thinking but I've been receiving some push back saying that TINs are not required for Non-Resident Aliens if they are not engaged in a trade or business in the U.S. and that is proved by the W-8 form.

TINs, by law, are not required for NRAs. They can have a TIN, but if your policy allows, they don't have to have a TIN if they have any of the four things WyoRockies stated above.

In your opinion, does the W-8 form qualify as one of the four items listed by WyoRockies? (I have never seen a W-8 form.)

CIP requires an identifying number as one of the 4 pieces of information that you are required to obtain. Having the customer sign a W8 certifying that they are not a US person subject to US taxes is 1 thing, but it has nothing to do with CIP. Your policy should state what is required for a NRA (most likely a passport).

I agree with devsfan. I apologize for any confusion on my previous answer. The TIN is one piece of information that a non US person might, but not always have but for identification purposes I would want the following:

Under normal circumstances, the bank shall examine an identification document evidencing the nationality or residence of the individual and containing a photograph or similar safeguard. : Acceptable identification documents would include an unexpired government-issued driver’s license, passport, or an unexpired government-issued identification card.