Making Delayed Charges Postable Transactions Possible?

As a small service oriented business in the education industry (tutoring, specifically) who invoices clients on account at the end of 2 week billing cycles, I need to conform to the accrual basis of accounting.  The problem I am having is with the need to invent or develop workarounds to record accrued/unbilled revenue.  Since I invoice every two weeks, and complete my basic accounting reports at month end, I will almost always have accrued revenue to report on each balance sheet (and, simultaneously, revenue on the P&L that hasn't been invoiced yet).  This is because the month end rarely lines up exactly with the end of my billing cycle.  So, for example, I have a billing cycle running from Sat. 5/21 to Fri. 6/3, and I invoice all those jobs on Fri. 6/3.  Well, most of that revenue was earned in May and needs to be reported in the May monthly accounting period, even though it wasn't billed in May.  I hope this makes sense.

Well, I got excited when I discovered the Delayed Charge feature in QBO, until I realized that they are non-posting transactions :(

It would be really nice if I could adjust a setting to make them postable transactions, specifically to Accrued Revenue on the balance sheet (debit) and Revenue on the Profit and Loss (credit).  Then, as I generate invoices from these Delayed Charges, Accrued Revenue would be credited and A/R would be debited by the amount on each invoice created from a Delayed Charge.

I don't hold out a ton of hope here, but is this possible to do?  I know some of you are thinking that I should just change my billing cycle to be monthly, but I have a billing rhythm as the business owner and my customers have come to accept that rhythm over the years.