I don't think it will soften to the point of flowing back out. I have used Elmer's Wood Stabilizer that is water resistant. It dries clear but in the rain it turns white again, yet stays hard. Then back to clear when it dries off. I suspect this would happen with water resistant wood glue? Plus the hole is deep. I am having to layer it on. So hopefully, if it were to get wet, the layers below won't be affected?
I don't want the wood to get wet at all.