Looking at section 5 of the form 656, I see this:
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Electronic Federal Tax Payment System (EFTPS)
List offer payments made through the Electronic Federal Tax Payment System (EFTPS) below.
Include the 15 digit Electronic Funds Transfer (EFT) Number with each payment.
Offer application fee Date EFT number
Offer payment Date EFT number
Note: Any Offer Application Fee or initial payment made via EFTPS must be made the same date your offer is mailed.
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So I think this indeed gives me path that I am looking for. On day I mail the OIC I will make the up front large payment via EFTPS despite submitting as low income tax payer.
