Hi, Dave,
I think you are going to an awful lot of extra work in cutting the wood into strips and alternating the grain to begin with.
If you think the extra moisture from the glue is part of the problem, perhaps you can just eliminate the sawing and glue up.
When glueing the backer onto the plaque , it really takes a lot less glue than most people realize. Perhaps you can
ease up on the amount of glue you are using there and, if you are still concerned about the backer not adhering well
enough, with 5/8" stock you can easily add a few brads.
If you do want to finish the wood on both sides, I think the glue will still hold just fine unless you are using an oil based
finish and, if you add the brads mentioned, that will eliminate that worry also.
You certainly do amazing work and your attention to detail is commendable.
Good luck and God Bless! Spirithorse