I think its probably got to the point where the only thing to really do is some temperature testing. I don't have much in the way of equipment to get things to precise temperatures, let alone to keep them there when in the breadboard, but...
I'll take a particular pair of transistors, see how they peform in this application at room temperature, then let them heat sitting on a 3d printer bed for a while and try again, then pop them in the fridge for a few hours and another try. That gives a far larger temperature range than I'll actually need the finished peak detector to work for, so it will show an extreme limit of how bad temperature variation can be. Maybe it won't be so bad afterall, I might be over-worrying about these sorts of effects simply becaue I haven't done anything analogue and precise with discrete transistors before, it just I see temperature and manufacturing variation effects mentioned in the context of discretes where you never hear them warned of for op amps or digital chips. If it is enough to be troublesome, calibration is the way to go I guess, I can think of an easy way to make it automatic* and provide a dummy channel for reference which can give datapoints for a signal of known size and for a DC state.
*I'm not mass producing these, but I am making a sort of module which can be included in a variety of future projects and I like to ensure any I build can be quickly put in to use and not need huge amounts of probing for individual calibration, but I think there might be a nice way to make it self-calibrating.