@Thomas Visaggio
Each housing authority is a little different but generally your understanding is correct. Essentially, you are renting the house as a conventional unit and converting it to a voucher unit. However, I wouldn’t recommend it as it creates a couple of complication and potential problems.
1) your HAP contract start date wouldn’t match your lease date, which is an issue for the authorities I work with.
2) here you can only get so many inspections before you have to wait a period of time. If you fail again you may end up in a situation where you can’t enter a HAP contract and then have a tenant that can’t afford the full rent and likely has to move to keep their voucher.
3) if you convert a unit from conventional to voucher you cannot raise the rent if the voucher would pay you more. In your scenario if you only prorate the tenant their portion and not the full contract (lease) rent and the authority found out, then the authority could claim that the rent for the unit is lower than your claiming and redo your HAP contract to a lower amount.
Most authorities have a landlord liaison position or something similar. That person would be the best to ask for your specific authority.