My first thought is to ask, what is at stake in Joseph's participation in the creation of the GAEL that would not also be at stake in his participation in the EA?
Joseph wrote "Egyptian Alphabet, circa Early July–circa November 1835–A" by his own hand, demonstrating his involvement in the EA project. Given that the methodology of the GAEL and the EA appear to be largely the same, and that their contents overlap and intertwine, whatever is at stake in Joseph's participation in the GAEL is also at stake in his participation in the EA--which is demonstrable. So unless we are going to throw Joseph out for participating in the creation of the EA, it makes little sense to worry about his apparent participation in the creation of the GAEL. Nothing is at stake in the one that was not already at stake in the other, and few of us appear to have left the church simply because Joseph's hand appears in the EA.
But that just gets at what is (or is not) at stake in Joseph's participation in producing the GAEL. What are the evidences that he actually did participate?
Here are a few. (By the way, for the full arguments on the below and documentation, see Don Bradley and Mark Ashurst-McGee, "'President Joseph Has Translated a Portion': Joseph Smith and the Mistranslation of the Kinderhook Plates," in Producing Ancient Scripture: Joseph Smith's Translation Projects in the Development of Mormon Christianity (University of Utah Press, 2020).
First, the conjecture that the Grammar and Alphabet of the Egyptian Language in the handwriting of Joseph Smith's scribes was a creation of those scribes has always been a poor explanation. Given Joseph's status as the translator of the Book of Mormon and Book of Abraham, he is the default translator for related works, particularly when the others involved in the related work were his scribes, and writing things down for him is, by definition, what his scribes are tasked to do. And Joseph's clear participation in the EA project further suggests his participation in the closely related GAEL. These considerations collectively make Joseph's participation in the GAEL project the default working hypothesis, which can then be strengthened or weakened by further lines of evidence.
Second, Joseph's use of the GAEL to translate from the Kinderhook plates indicates that he gave credence to it and raises the question of why he gave credence to it. It is unclear why Joseph would have given the GAEL sufficient credence to use it as a translation tool if it were simply composed by W. W. Phelps and Warren Parrish, but it is clear why he would have given it such credence if he had been involved in working it out and it therefore represented his thinking. We all, axiomatically, believe our own beliefs, but those of others, not so much--at least not without good reason.
Third, the eye-witness account of Joseph comparing the Kinderhook plates to the GAEL says, of Joseph's engagement with the characters on those plates, "He compared them, in my presence, with his Egyptian alphabet…and they are evidently the same characters. He therefore will be able to decipher them." (To read the original letter, as published in the New York Herald, see column 3, here.) The author of this letter, Nauvoo judge Sylvester Emmons, as a non-Mormon mistakenly connected this "Egyptian alphabet" with the Book of Mormon rather than the lesser known Book of Abraham, but it is evidently a reference to the GAEL, which bears the title "Egyptian Alphabet" on its spine and is identifiably the source of Joseph's translation of the Kinderhook plates character known in the GAEL as "ho-e-oop-hah." A key phrase to observe here from Emmons regarding the GAEL is that the letter identifies the GAEL as "his Egyptian Alphabet," suggesting that Joseph represented himself as the one who had derived the GAEL's contents.
So, bringing together all these lines of evidence that Joseph took part in producing the GAEL, we find that 1) Joseph was the default translator for the church and on the Egyptian papyri, 2) Joseph demonstrably participated in the GAEL's parallel and overlapping project, the EA, 3) the GAEL is recorded by the hand of Joseph's scribes, whose duty as scribes (what it meant for them to be his scribes) was to record text for Joseph, 4) while Joseph would have presumably believed what was in the GAEL if he helped produce it, there is no obvious reason for him to have relied on it if it were the sole product of his scribes, 5) yet Joseph does rely on it to translate from the Kinderhook plates, with 6) an eye-witness reporting him representing it as "his Egyptian alphabet."
Could one hold that Joseph Smith did not participate in the creation of the GAEL? Certainly, but only if one wants to bet on side of probably being wrong on the issue even though nothing new is at stake on it that wasn't already at stake in the issue of EA, and that wasn't already laid to rest with the issue of the EA when that text was found to be written out in Joseph's hand.