"put X down to" vs. "put down X to": subjects of verbs with two particles

Solution 1:

In this case, the phrasal verb is put down as you have correctly said that the particle can go either side of its direct object

to put down many coats
to put many coats down

The to immediately following that phrase is part of to do the job.

In order to completely hide the horrible colour, I expect I would have to put down many coats of paint.

Now, there is an idiom to put down to, as in "I put the high cost of food down to the price of fuel" which means "I believe the reason for the high cost of food is the price of fuel". Your original example doesn't use this, and it's not a separable phrasal verb (as in your Biber reference) because down to can only come after the direct object. You can tell the difference because the idiomatic put down to is always followed by a noun or noun phrase.

Solution 2:

There is no misreading possible, simply because “to put something down to something” doesn’t allow the down to be separated from the to. These are two completely different senses, of which only the last is an idiomatic phrasal verb with a special meaning:

• I put the whole thing down the drain to empty out the pail.
• I put the whole thing down to get a fresh start on it in the morning.
• I put the whole thing down to poor management.

I don’t recognize any of the rest of the question as making any sense. It’s too much hyper-analysis, the kind of stuff they needlessly saddle ESL learners with, making them think it is a real thing, and really important, when it is not. The proof is that nobody ever teaches this sort of silliness to native speakers at grammar school.