This is a really interesting question both as a freemason and someone with an interest in history. After seeing it I started some research and, to the best of my knowledge, there is very little information on this seemingly glaring contradiction. Although only nine of the men who signed the Declaration of Independence are thought to be masons, that included Washington, Franklin, and Hancock, all hugely influential figures in the formation of the Republic. Because of this, it made me think what motives the founders of the Anti-Masonic Party had that would allow them to gloss over this conflict.
The best primary source is the organiser of the party, Solomon Southwick- a fierce supporter of the Revolution and son to a printer who actually communicated directly with Washington to print some of the copies of the Declaration. From what I can find, he is the only prominent leader of the Anti-Masons who addresses the conundrum with the Masonic influence over some of the Founding Fathers. To do this, he essentially lies about what Washington said or purposely misconstrues his words. Southwick asked his readers,
"Could the patriarch (Washington)...who had drank deep at the fountain of light which emanated from the divine mission of the redeemer of mankind... slake his thirst for improvement in the scientific and moral deserts which comprise the domain of Freemasonry? I boldly answer no, he could not, did not".
To back up his claims, Southwick argued at Washington's farewell address he warned the country of secret societies that would "subvert the power of the people". The problem is, there is zero evidence Washington ever said this. This was certainly a feeble attempt at a cover-up but in Southwick's eyes clearly aimed to pardon Washington of any masonic 'crime' meaning his motive and venom were directed elsewhere. The overtly religious tones of the argument (indicating Washington's mission had divine backing) fit in very well with the evangelical support the Anti-Masonic Party had particularly in rural North-East America. On top of this, and most importantly, Southwick and his supporters were undergoing a political realignment that trumped the masonic connections of their country's founders (or at least some of them).
A strong example of this is John Quincy Adam's support of the party. Adams had received political appointments from George Washington and even named his firstborn son after him- it doesn't appear Washington's masonic connections blinded Adam's into disliking him at all. From this, it begs the question as to why join the Anti-Masonic Party which on paper would indict many of the Republics' founders? It would appear that the Andrew Jackson divide in politics drove Adams into the hands of the Anti-Masons which was used as a tool against the former Grandmaster of Tennessee. Alongside his fervent religious beliefs, Southwick too was feeling disconnected from his former Democrat-Republican beliefs and looked for alternatives, driving him towards the growing religio-societal discontent with Masons.
Ultimately, the Anti-Masonic Party grew from the political divide and coalesced with strong religious opposition to masonic ideals and beliefs. These more pressing issues at the time allowed glaring contradictions to be either ignored or poorly papered over, in the case of Southwick, and directed instead towards the thorough dislike of President Jackson.
Edit: If anyone can find more primary sources addressing the question or other titbits I’d be very interested to hear.
Taylor AM, Young Charles Sumner and the Legacy of the American Enlightenment 1811-1851 (New York, 2001)
Stein M, American Panic: A History of Who Scares Us and Why (New York, 2014)
Bemis SF, 'John Quincy Adams and George Washington', Proceedings of the Massachusetts Historical Society, Third Series Vol 67 (1944)
Bearden S, 'A Question of Timing: Anti-Masonic Hysteria, 1820-1850', (Unpublished Thesis, Ouachita Baptist University 1985)