This is the rule rather than exception among good quality current melodica models. Yamaha Pianicas and metal tray Suzuki Melodions (e.g., the M-32C) are especially well suited for your project, in my view, because of their solid construction inside and out. Another consideration, though, is the spit valve mechanism, which definitely has to be adapted to the wooden casing. This may be easier to do with one melodica than another, depending on your design. A notorious defect of the most well known wood-cased melodica, the Mylodica, is that it (either the alto or soprano model) has no moisture vent, which creates a problem for the wood and more generally. I have the soprano Mylodica, which contains the works of a Suzuki S-32C, but the vent exit is simply sealed off.
The Mylodica is also bulky and angular, as wooden “cigar box” melodicas typically are. It is highly desirable to make the wooden case as trim and comfortable to be held in the left hand as is feasible within the limits allowed by other design imperatives. The Banarsë and Bruno’s wooden P-37D incorporate wood into their design without bulk and (it appears) undue weight. I can’t tell for sure, but the tray of Bruno’s model looks like the original plastic rather than wood.