If you are playing without effects, and if you are trying to use a amp as “part of your instrument,” contributing to the sound in a significant way (very much like guitarists and harp players do), then I could see why you may be having trouble finding an amp that compliments your various melodicas.
I gave up on keyboard amps (for my organ and other keyboards) about a decade ago. I found, like you have, that they had too much character of their own, and from my perspective, were kind of cheaply built. The sound from these amps at volume was horribly bright and brash and often distorted. So I started using separate components (power amps, crossovers, and high-quality non-pwered speakers) or high quality monitors in place of the keyboard amps. I shape my sound before it gets to the ultra-clean, ultra-flat amplification system. Warm sound in equals accurate, warm sound out. I have some 10-year-old JBL, bi-amped, two-way, 10-inch monitors that sound about a smooth as stereo speakers, and these are my “go to” personal monitors (and speakers in small venues). These are the best sounding monitors I have heard to date. I just ensure the sound going into them is good, primarily by setting up the sound using noise-cancelling headphones first.
If you are using your pedals to shape you sound, then again, a high fidelity amplification system should reproduce the sound you feed it. If it sounds good with headphones, then the amp should reproduce the sound. Like Binyomin, I often have difficulty ensuring my sound is right because the acoustic sound off of the melodica is loud and disguises the sound coming from my monitor. I tried some in-ear monitors, but I returned the ones I purchased because they sounded to tinny and were hard to use. I am still exploring that avenue (headphone monitoring) with noise cancelling headphones.
The only thing I can recommend that may be a good option is the modelling guitar amps. I think Line 6 makes one. Several companies make them. These are typically solid state amps that have the capability of modeling all the classic amps ever created. I am not biting, but that could be an options to try out. I am awfully tempted to try the Bose L1 system. You can purchase it with a T1 Tone match processor that offer 100 EQ settings to enhance known microphones and room setups.
I struggle with your problem too.