Welcome to the Wefugees community and thank you for your question!
Just a heads up: I'm not a lawyer, so all I'm sharing with you here is what I've come up with during my research.
As you already mentioned, due to his illegal status he can't travel to the U.S. - and since he is illegally in Lebanon, I'm unsure if that is an option (the Lebanese marriage law seems very strict and if neither of you is Lebanese, it could be impossible to get married).
Here is what the U.S. Embassy Lebanon says on marriage: https://lb.usembassy.gov/u-s-citizen-services/local-resources-of-u-s-citizens/marriage/
Also, an overview by the British Embassy Beirut: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/701258/Overview_about_the_Lebanese_law_of_marriage_November_2017.pdf
I'm sorry that I can't help you more. Depending on where you are in the U.S., it may help to reach out to a refugee support service and ask them for advice! Here are the UNHCR - USA Contact details: https://www.unhcr.org/en-us/contact-us.html
Feel free to reach out to us again if you need any more assistance or to share an update on your situation with us.
All the best,