I do not think you would be happy using laser for your application, i beleve a CNC router would be the ticket.
About two years ago i built a CNC router for the purpose of doing inlay work, mainly with abalone, Ablam, MOP, gold sheet, and etc… into hardwoods. This was musical istrument related work, so quite fine in detail.
I chose a build it yourself kit, basically made from speciality extrusions and some machine specific parts. I used good stepper motors and a four axis controller. I spent about £1500 for a complete 750x1000mm bed router including the motor and a good assortment of tooling.
For a supplier I choose to go the OOZNEST open CNC route, however there are several good suppliers of do it yourself CNC in the UK to pick from.
Check out this web address for some info: https://ooznest.co.uk/product/ox-cnc-full-kit/
I was able to hold very tight tolorances, far better than I ever could with gouges and a Dremel tool. It was amazing that a belt driven X and Y axis with ball screw Z axis machine was that “Tight”.
At first I was quite skeptical, thinking that I needed ball and screw drives to maintain a better than .015" tolorance but it turns out that is not the case. The material in the belts is made with some sort of carbon fiber in its core that just does not stretch. As long as you have a tight fit in the pulleys you will experience almost zero backlash. Of course you can dial in a backlash factor in your sofware if you wish, but i found it was not at all necessary for simple inlay work.
I would not try to hold .001 tolorances with that type of system, however for fine inlay and marquetry work it is perfect.
What was even better I was able to use my V-Carve Pro software that has the ability to generate inlay specific G-code files. that is it created both the inlay and the pocket toolpaths from a single drawing! A really big time saver.
With that software i was also able to simply import an image of a pattern I liked and it would do abut 90% of the graphics work for me, I would only have to clean it up a bit before generating my tool paths.
It was fun to build the machine. There is an active community that you can chat with acessable from the OOZNEST site. The same goes for V-Carve Pro. The V-Carve people are also UK based here is a link to their web site: https://www.vectric.com/products/vcarve.htm
Good luck with your journey into CNC, I think you will find it quite enjoyable.
Using CNC for inlay work shifts a lot of the inlay experience from endless hours with a jewelers saw and Dremel tool to being creative, concentrating on the design whilst the machine takes care of the drudge.