Tru-oil changing colour of stain

[McCreed]

The "Tru Oil turns blue to green phenomenon" is well documented here and other forums.
So to answer your question: "Will I have the same problem with a royal blue stain?" The answer is yes.

As for other clear coats that will maintain your original base colour, there is polyurethane, or lacquers. Both will stay clear without affecting the base.
I presume your choice of Tru Oil was at least partially motivated by the fact that you can apply it by hand, not spraying.
If that's correct and hand applying is your only option, I would suggest poly. You can buy "wipe-on" poly or you can mix your own with oil based poly.

Both polyurethane and lacquers (acrylic or nitrocellulose lacquer) can be purchased in spray cans providing you have the space to carry that out. Rattle can spraying can certainly be done outdoors, then the guitar brought indoors to dry, so you don't need a fancy spray booth or set up. However lacquer fumes can be quite strong when gassing off in your house/apartment.

I was also wondering if I could stain the veneer black then sand it back and then stain it blue but I’m not sure if that will be a bad idea with the veneer.

You can use black stain/dye then sand back to enhance the grain, then apply your main colour. There have a few threads here recently on the subject. The forum search engine is shit, but if you look around you should find something.
You do need to be conscientious of sanding the veneer as it is only about .6mm thick, but completely workable. It's about using common sense more than anything. It's not rice paper.