I'd adjust the end of the neck as you suggest.

You can steam the veneer, it's always worth a go. I'd use the soldering iron and damp cotton rag method rather than a larger iron.

You can do a light sand of the veneer afterwards, but steam first as it will reduce the amount of sanding required. Let the wood dry out thoroughly after steaming before sanding.

Be careful when staining as the binding has small stress cracks in it that you'll only see once stain gets in them. The tighter the binding has been bent, the more cracks it gets. So those two front 'ears' are full of the things! Aways mask off the binding if staining. I've found that spirit-based stains penetrates the binding far more than water-based stains do.

For colour there is always the spray option with a heritage cherry red tinted lacquer spray. You'll probably have to go to nitro lacquer for that as getting tinted clear poly or acrylic is quite hard. Someone found a source of one of those types, but they aren't common. If you have a compressor and spray gun, you could probably mix your own poly or acrylic tinted lacquer though. Nitro spray cans aren't cheap, and it's quite a big body, so you'd probably need a couple of cans of the red and a couple of clear lacquer to go over the top. Spray enough coats of the red until the finish is deep/dark enough for you, then put protective clear lacquer over the top of that.

Lacquer has the benefit of not really being affected by glue spots as the colour is sitting on top of the veneer, not being absorbed by it (or not). Again, best to mask off the binding (3mm vinyl auto pin-stripe masking tape is great for following the bends), but lacquer can be scraped off the binding. Red lacquer on the body wood only, but I'd clear lacquer over the wood and binding.

There are any number of other staining options available. Leather dyes or Feast-Watson stains are often used by people on here. They are really all very similar despite being marketed for different end purposes; a pigment in a carrier liquid, which may be water, spirit/alcohol or oil based.