A shame you've got a few issues with the kit, but you seem to have a handle on it. Very nice pattern on the veneer.

I can see why you want to darken the wood, you just need to be careful not to over-darken it so you loose the full spalt effect. So I'd suggest using a quite diluted stain, and go for more applications and slowly build it up rather than trying to do it in one go and then get too dark a result.

As the veneer top is very thin, and getting it too damp can cause it to become unstuck in places, I'd try and use a non-water based stain. Anything you can dilute with turps, methylated or white spirit should be fine.