The new covers would certainly make a difference although I personally don't find the CB necks over warm in the slightest. In fact they are the lowest wind of all our Tele necks and have a reputation for being extremely clear. It's worth checking your pots and caps to see everything is as should be and another quick check is to wire the neck coil straight to the jack socket so you're hearing the coil without the effect of any of the controls-at least that way you can rule out their effect.Removing the cover is another option and will make a considerable difference in tone.
I'm probably being harsh, but the tone of the neck doesn't balance at all with the tone of the bridge not like I would expect a calibrated set to.
Trying this pickup set through my new fender champ 600 on full volume and controling the output with the guitars volume only makes this more obvious. I think I will try and wire the pickup directly to the jack and see what happens, but I'm unsure why this may be a factor as the bridge shares the vol, tone and cap with the neck.
Although it's by no means a bad pickup is just doesn't balance with the bridge. I'm starting to get concerned that the neck PUP may not be a genuine BK, as there are no markings to suggest it is unlike the bridge which is clearly stamped as a BKP.