They do make
low odor water base contact cement. Weldwood has always been a leading brand in my opinion.
I never use the water base, but it should be perfect for your application.
If you wanted to use it, I could give you some tips with pictures so you could stick it perfect first time. Let me know if you change your mind.
It's definitely easier than cutting strips and trying to put them back together.
Figuring out how to do something you have never done is what makes a good challenge.