I was a finisher till about a year ago, you've got a few options, hopefully I can explain them well enough.
First of all, if you have any of the material scrap, use that as a test for anything you do. Run the colors on that scrap by the client to get approval.
Option 1: torch, I do not know enough about this to really be helpful other than: you better have a precise flame and a steady hand.
Option 2: Stain. Some woods require a pre-stain so that the color doesn't end up blotchy, do your research and figure out if this is one of them. I have a major preference of minwax stains, try out a few colors on your scrap. Practice brush work precision, like if you're only hitting the branches tips of the trees, make sure you can brush stain on a small of an area with accuracy. Make sure that in wiping the stain off it doesn't bleed to areas you do not want it. Again, practice on some scrap. Seal it (after the staining), scuff the sealer, clear coat it. I recommend Campbells Krystal conversion Varnish.
Option 3: Airbrushing. This has been mentioned a lot. Its a good method, you still need a steady hand though and I would recommend masking off areas you are not working on, aka work in sections. Again, if you have scrap, try it on that first. Seal your material first, then airbrush with a toner (clear varnish mixed with pigment, very little pigment, the mix should be almost transparent so you can layer up to the opacity you want). Mohawk sells cans of toner in a lot of different colors, it works in a pinch, I just personally think spray can nozzles are harder to control. With this method, you can fix it if you over apply by sanding off the colorant, and even if you oversand off the colorant and end up hitting the sealer, you can just reseal and try again. Sealer, scuff 400 grit or higher, toner, clear coat. Krystal is still highly recommended.
Option 4: Touch up sticks. This is a similar method to air brushing, but instead of spraying, you'll be using Mohawk touch up sticks. Seal your material, scuff, color with touch up sticks, coat in clear coat. These sticks come in many colors and work like crayons, you can rub in the colors with your fingers, the heat of your hand and friction will make the colors blend. I have never done this on a large scale but it is something I have used to correct stain colors on sanded through veneer or on wood filer that wouldn't take stain and I do not see why it would not work here.
Best of luck to you. Please try on scrap before applying to your finished sign.