There are latches which have a HDPE strip on the face. I have no idea whether they can be purchased separately as a retrofit, or whether that would reduce the sound significantly. You'd still have the sound of the latch springing back into the strike opening, though it might be possible to muffle that.
There are latches which swing back into their recess rather than being "wedged" inward by a sloped face. I've only seen these on fairly high-end mortise locks; I don't know whether they'd be available for anything else. (They can have other advantages.) Since they operate a bit more smoothly they might reduce the latch noise.
The only suggestion I can offer that I have some real confidence in is to get rid of the mechanical latch and strike, replacing them with something that engages only after the door is actually closed. A magnetic locking system, for example, can engage without an audible click when properly installed. Of course then you need power supply and control circuit to handle locking/unlocking it, and there may be life-safety requirements for exactly how that's installed and operated.
Of course the simplest and least expensive solution is to stop worrying about this click and learn to ignore it. Seriously, unless you have VERY weird requirements, you're overthinking this by about 1000%.