I hate doing this but you might want to get some locking nut shims. Why I hate doing this is because it is trial and error to find the right balance. I have a strat that the D string was doing the exact thing. Had to add one shim, tune it back up, add another shim and then play with the relief +/- and do this process a few times. Most of the times adjusting the Floyd height or playing with the neck relief is the key but sometimes the nut is too low. Also sometimes you need to shim the neck itself to either angle it downward or angle it upward. Sounds like you said that you already have the relief really loose and maybe you need to mechanical shim the nut or the neck and then you can tighten the truss rod back up to compensate the other way for the mechanical shimming. I know this is a huge headache with Floyd equipped guitars but at times it is a necessary evil. Most guitars that have given me grief with singular problems like this end up being the sweetest sounding guitars in the end because from sheer determination and trial/error you have gotten the string height and neck relief just perfect and the harmonics just jump out.
Another area is the spring tension on your actual Floyd. I block all mine so they don't float but having the springs too tight on a blocked Floyd also affect your tone and feel of the strings.
Sorry for all of the options to look at but I speak unfortunately from dumb luck myself and have experienced most of these anomalies. Robert