Does anyone have rule of thumb cost estimates for this kind of work?

There's a very good chance the issue(s) you face have already been discussed and answered.Ask questions in the appropriate forum.

Your chances of winning are close to zero.

I was wondering if anyone would be able to give advice on if there was a chance of getting a HELOC in my current situation.  

This not only lets your get a discount when these suppliers compete against each other, but if you have detailed plans from an architect or a spec for the build out - they can also estimate all the materials needed for you.

This why I can so easily say NO, if it were '08' all over again NO, 99% of you would NOT be buying like you lament and promise if ya just had a chance.

Assuming I buy a single-family home in Washington as a primary residence (better chance of being approved than a Multi-family).

Conservatively estimating new construction costs as $300/sq foot, the new duplexes could cost about $600,000 each to build (plus interest costs).

Photos will not work on estimating.

Please ask your question once and give people a chance to answer.Call the Housing Authority and ask them.