For such applications, I typically use a common PVA glue like Titebond II. I believe Carole Rothman prefers Weldbond when gluing up layers for her stacked ring bowls. It's a PVA as well. Both dry clear and should not present a visible glue line. Interfering with finishes is another issue. Any PVA glue residue that isn't removed form the surface of the wood, will show up noticeably once a finish is applied. It's important to inspect any glue up for this prior to finishing. You could try a hide glue. Titebond offers a pre-mixed version of hide glue. It supposedly does not interfere with finishes, but I'm not certain if that means that squeeze out won't show under a finish or not. Hide glue is a dark glue, so the glue line may be visible, especially contrasted against a lighter colored wood. I recommend testing it on some scrap before using it on a project.
I really think you will be fine with the PVA glue of your choice. Even a white PVA, such as Elmers glue should work well in your application.
