If you're really going for high levels of detail, I doubt that you'll be happy with the results of FDM printing. By its nature, it will have fine grooves over the whole surface where the individual layers of melted plastic adhere to one another. I'd suggest holding off on your purchase for now, and send some samples out to be printed on various different types of machine. Go through your proposed workflow and see how well each of them works for you. If one printer does produce results superior to all the rest, you'll know which one to buy. Include a resin printer in your trials, like the Form2. These, while they have their problems, will produce better surface detail without the pronounced lamination lines, and don't need the acetone smoothing referred to above, which will melt away fine details.

The suggestion of CNC routing was a good one; the parts you're talking about seem like things that would be well-suited to that process, which can produce very clean surfaces. (This sort of part is not what's usually referred to as 2.5D, by the way - that's something that's a series of flat planes at different heights. I think the term you're looking for is "reliefs".)

You objected to the router idea because of undercuts, but realize that in most 3D printing processes those undercuts won't come out cleanly because they'll be filled with support structures that will need to be removed by hand and cleaned up. If there's not too many of these undercut areas, it would probably be just as easy to carve them in by hand after doing most of the work with a 3-axis router. If not, use a 5-axis router, which can carve them cleanly. Or use a SLS printer that doesn't require support structures. Neither of these machines are going to fall within your budget, though.