Learn English – When can we change the order of the particle and the verb in a phrasal verb

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.

Some rules of thumb:

• if the preposition indicates a genuine "direction" or essentially carries its 'intrinsic value' as a preposition, then it comes before the noun or pronoun as you would expect with a preposition. For example: He turned off the motorway. He ran out the room. Notice how in these cases, you can often take away the prepositional phrase and the sentence still more or less makes sense, though is much more vague ("He turned", "He ran").
• If the preposition is effectively "fused" into a transitive verb, and especially where the choice of preposition is fairly arbitrary, then it is likely that will come before a noun (but after a pronoun). He carried out the experiment. He carried it out. Contrast: He carried the boxes out, when out has its more literal meaning, and notice how "He carried" bears no relationship at all to the sense with 'experiment', but bears some relationship with the sense of carrying boxes.
• With some prepositions that have a "semi-figurative" sense but still keeps some of its value of the preposition, you can put the preposition before or after a noun, arguably depending on whether you want to the emphasise 'directionality' of the preposition: He turned off the tap or He turned the tap off.
• A "figurative" preposition fused into a transitive phrasal verb is in any case likely to come before the noun phrase if it is relatively long. So for example: He turned the taps off would be common, but He turned off [all of the taps in the house as well as the house next door] would be a common word order because of the length of the object phrase in brackets.
• If you coordinate prepositions that are fused into a phrasal verb, then they usually go after the nouns. For example, He turned the taps off and the washing machine on is fine, but you can't generally say *He turned off the taps and on the washing machine, even though individually you can say either He turned off the taps or He turned on the washing machine. (With "genuine" prepositions, coordination works as expected: He turned off the motorway and onto the ring road.)