I am actually considering doing just that, however, the instructions for Form 7004 state: "The extension will be granted if you complete Form 7004 properly, make a proper estimate of the tax (if applicable), file Form 7004 by the due date of the return for which the extension is requested, and pay any tax that is due."  (Emphasis added).   The instructions do say later "A trust (Form 1041) or REMIC (Form 1066) will be granted an extension even if it cannot pay the full amount shown on line 8."  Does this apply to an estate?   Would not paying the $1 actually invalidate the extension if there is an estate instead of a trust?  Who knows.  Is it worth the risk in the event tax does end up being due?

 A much better solution which avoids all the uncertainty, the trouble of paying the $1, or certified mail costs to paper file is for ProSeries to update their software to remove the requirement of the balance due being more than $0 in order to efile the 7004 when there appears to be no IRS requirement supporting it.  Requiring a non-zero balance is a recent change to their software.