Some LVL's are coated with a wax. The ones I just used had a yellow coating. The construction adhesives will not stick to them. As the others have said, screwing or nailing them together will be fine.
Where did you get those lights??
I don't comment much but I have been following your thread for a couple of years. Incredible, simply incredible!!! I'm borrowing your sunshade idea for my pergola. Sometimes the sun is just too much, even in Chicago.
I have the New Air G73 installed in a 2 car garage here in Chicago and it has been awesome. I know of 3 other other people that have the same unit and they are very pleased also.