That stuff isn’t all that good around wet areas like a bathroom. You’d have to seal every joint. If water gets in there, it’ll soak into the particle-board underside and all the edges curl up.
Friend of mine’s kitchen floor was ruined after a kitchen sink overflow.
ETA: I see that’s vinyl. Nevermind - not the same stuff…
Thanks again, all, for your good answers and ideas. If this thread doesn’t sink too far before I get a quote, I’ll let you know what dude says.
Before using lauan, ensure that it is stamped “underlayment grade”, otherwise, it’s not acceptable per resilient flooring institute standards. For a bathroom, 1/4" Hardibacker or Eternit Ultraboard is/are the product to use. Likewise, using ordinary water based flash patching compounds are a bad idea. Not only do they take a while to dry, environmental moisture can cause them to become unstable in the future. Ardex SD-F is the product for all underlayment feather patching applications.
Yes, these are more expensive products, but I don’t make money on call backs or poor customer satisfaction. 
Sorry I didn’t mention the grade of luann, but after talking with Armstrong vetted floor people what it boiled down to was being sure exterior glue was used.
All grades available to me use X glue.
The floor patch was supplied by the tile vendor but I don’t recall it taking long to dry. Despite fifteen years of raising children (sauna style half hour showers, stepping out wet and puddling, occasional overflowing toilet and the like) the floor itself remains trouble free.
I could sing the praises of acrylic lacquer on the woodwork holding up under the sauna conditions too.
Just to follow up on this…
I ended up getting a pro carpenter to come out and do this, because none of the FOAF folks were calling me back. It cost me $1600, plus I bought my own tile and grout.
This is what he did:
I think it was a great price to have a pro job done. I thought I’d let everyone know the price in case anyone else has to embark on such a project. Thanks for the advice on the flooring, too. I am very happy I went with professionally-installed tile!
Once again I’m too late for the party, but bit about rotted boards leaped out at me. Did you have them check for dry rot or wet rot?
The floor was rotted around the toilet, because the toilet tank had been leaking before I bought the house and no one fixed it. So the floor rotted and the seal around the toilet was bad (due to the rotten floor) so the toilet leaked more and rotted the floor more, and on and on.
When he came out for an estimate he told me that there was a chance that the joists were rotten and if so, it’d be a bit of a new project and a new price to replace (or fix?) them.
Turns out the floor joists were ok. Most of the floor was ok, just not the part near the toilet. So replacing the rotten floor and sealing everything up well, and adding some cross-beam supports between the joists, will keep me nice and dry and free from further rot.
Is that what you mean?