I've used water-based stains on veneered basswood. You aren't soaking the wood enough to loosen any glue when staining, or you certainly shouldn't be. Let the wood dry properly after each application of stain so you can see the final colour.

Better to use too diluted a stain to start with than too concentrated. You can always stain again, or move up to a more concentrated stain if it remains too light.

The water might raise the grain a bit, but it's only loose wood fibres that pop up, and a lot of people will dampen the veneer anyway to get them to pop up so they can be removed. You only need a very light sand over the top. Stewart Macdonald in their guitar finishing book call this a 'drag-sand', as you just hold a piece of sandpaper by the edge and drag it over the surface a few times to pull the loose fibres off. You really aren't taking any depth off the veneer doing this.