That is totally normal & typical of relative output Strat to LP.  It's also exactly why clean boost pedals exist.  So you are on the right track, there!

It's also good that you've noticed the Strat is a more dynamic &, as you say, subtle instrument even with gain & volume.  You might also consider simply turning your LP down (with the guitar volume), instead of trying to push the Strat with pedals.  Of course these days with modern gear you can have multiple presets on digital or modelling amps & effect units to switch settings & output volumes at the push of a preset button as needed for whatever guitar you are playing!

I think what you are hearing with the fuzz pedal is also pretty typical.  Fuzz as an effect is a little odd in that it typically squashes any dynamics or upper harmonics by pushing the lows & mids to crazy clipping levels.  Most players I know that use fuzz (or a similar effect) also pair it with an overdrive or boost pedal to help tame the lows & push highs to get some kind of clarity out of it for upper range single string notes.  I'm inclined to say it will happen with any fuzz unit.

Another problem with fuzz & a Strat is if you are using the neck pickup it accentuates the bass range while cutting the highs making the problem worse!

Most problems of sustain can be solved with volume & a little more clarity in the mids & highs of your overall settings.

Hope that helps.  Enjoy your new Strat!