i have gigged and had band practices with a hot rod dlx many times. This is in a fairly loud playing 2 guitar rock band. The amp can produce very good tones, especially rock clean tones. Its interesting that your singer prefers neck pickup, muddy tone (as you describe) but you say the amp is a trebly mud, which seems a bit oxymoronic (jumbo shrimp, goverment intelligence, etc). I wouldn't start replacing tubes, speakers or pickups just yet, although all of those things will impact the tone. Instead you need to start by agreeing with him on what kind of tone is appropriate. If you're looking for a nice balanced clean humbucker tone then I would start with the guitar on the bridge pickup with volume and tone full on. Then go to the amp and set all the eq at 12 o'clock, slightly cutting the mid and boosting the treble. Of course you need to do this at playing volume because tone controls sound different at low volume than full band volume. Based on what you describe you may find this setting too bright. At that point you can slightly boost the bass, slightly cut the treble. If its still no good then try the both pickups togther, slightly back off the volume on the guitar or slightly back off the tone on the guitar. If you need to reduce either guitar volume or tone below 7 then its too much and you need to tweak at the amp.
If making these small incremental tweaks from 12 o'clock setting is still not doing it for you, the biggest bang for buck is trading in that LP studio for a Stratocaster. It will sound totally different but great for clean playing, and it will not take up as much sonic space as the LP. If budget is an issue the new Squire classic vibe strats are really impressive, better than the mexico strats in my opinion. Get a rosewood neck model and I suspect that will be the sound you are both looking to achieve.
Good luck and let us know what you try and how it works.
