Upgrade,
Our company does not insist (or even recommend) not doing a bleeds off take-off after de-icing. Usually the associated contaminated corrections will make a bleeds off take-off a necessity.
If your concern is fumes in the cabin, then the (737) manual says:
“If de-icing fluid has been used in an engine inlet, do not use bleed air for air conditioning from that engine until all fluid has been cleared from the engine.”
So why not start the APU up after the de-icing, keep its bleed air off, let it run for a while to get rid of the fluid and then use the APU bleed air.
If your concern is for the APU, then the same book says:
“With APU operating, ingestion of de-icing fluid will cause objectionable fumes and odors to enter the airplane. This may also cause erratic operation or damage to the APU.”
So we can never safely clear the ingested fluid without running and possibly damaging the APU – life is a bitch !
If you really need the engine bleeds off, and want to avoid fumes and a damaged APU, I think strictly speaking you are faced with an unpressurised take-off.
S & L