@Gary Parilis,
The delayed financing exception is what allows you to skip the 6-month seasoning period, and it requires that your purchase money isn't secured by the subject property. So, you can absolutely use an existing HELOC as your source since it's secured by a different property that you already own. And yes, the Refi lender will probably want to send money directly to the HELOC to pay it back.
Don't use private or hard money for long-term financing (not sure if this is what you were suggesting). You could potentially use private money or hard money for the purchase and then Refi with a traditional lender using the exception, as long as the private/hard source funds aren't secured by the subject property.
Another potential downside for the delayed financing exception is that you can only Refi what you paid for the property. Meaning, if you used a HELOC to fund both the purchase & rehab, when you Refi, it'll only cover the purchase, but the Rehab amount will still be outstanding. This also forces you to park extra equity in the property, which you may or may not be comfortable with.
If you want to Refi and get both purchase & rehab costs back, plus squeeze out some the added value, you'll have to wait 6 mo's.
You can definitely use the exception with BRRRR, as long as you're willing to keep a little more equity in each property. On the other hand, if you're an investor who wants to have max leverage on every property, the exception will work against you.