Well, OK, the problem of making the display visible is probably solved, but that brings with it another minor problem. My display has a touch interface, and I was vaguely planning to use that to trigger the display of some internal info or whatever for my own geeky needs. I was not planning to use that for reconfiguration or for actually controlling the lights, but I might be able to display some configuration info or logs or something.
I'm pretty sure the display I have uses resistive technology, in which case it definitely will not work through a plastic cover. Even if it turned out to be capacitive, the cover would probably have to be pretty thin for it to be usable. Oh, well.
I happen to have on hand a few APDS-9960 gesture sensor boards. They operate over I2C, and I already planned to have I2C for the BME280 environment sensor. I should probably be able to use one of those gesture sensors through the plastic cover for simple things like displaying extra "pages" of information.
I'll have to save additional details about that for when I get bogged down in software design.