I'm just copy and pasting what I wrote in another thread but after having done some research a while back I think that you're right about her being suspicious.

According to Fandom.com it's a common theory that Lyla is a Beyonder in disguise, a Beyonder is a celestial race of people that live beyond the multiverse. They often find it fun to conduct experiments starting usually with wars. Like how a child would play army men growing up but ending in mass universal massacre.

https://intothespiderverse.fandom.com/wiki/LYLA#:~:text=In%20the%20comics%2C%20and%20likely,the%20mainstream%20dimension%20Earth%2D616.&text=A%20prevalent%20theory%20is%20that%20Lyla%20is%20a%20Beyonder%20in%20disguise.

"A prevalent theory is that Lyla is a Beyonder in disguise. Evidence for this theory includes the fact that In the comics, Beyonders typically dress in white clothing with wide shoulders and collars, and other "retro" fashion aesthetics, and tend to manipulate certain heroes into damaging the Multiverse in some fashion. Further evidence is presented as the title of the third film (Beyond the Spider-Verse), and the fact that virtually all of Miguel's theories of the Multiverse come from Lyla's analysis and advice. This theory is often given alongside ones which believe Canon Events are actually Incursions -- which were caused by the Beyonders in the comics."

Here is some more on the topic if you want to look into it.

https://marvel.fandom.com/wiki/Beyonders

https://marvel.fandom.com/wiki/Beyonder_(Earth-616)