Boost before acoustic pre-amp / DI

I think codamedia nailed it. Your problem is too hot a signal with the boost engaged. Get your boosted signal down below the threshold where it thickens the preamp. Use your PA or powered speaker to make up for the difference in volume.

 

Getting more lightly played sections equal in volume to strumming is what compressors do. That’s probably their #1 use.
I don’t think you need a “boost” pedal to push the preamp of your amp ~ that fattened/altered tone is what boost pedals are designed to do. Unless you use something like the full sized Spark Booster with the gain way back (it’s technically a light overdrive, but can be very clean).

You should try the compressor you already have. Find a level of compression that lifts those lightly played parts up without sounding obviously squishy.

If you really have the itch to buy something, try a used Danelectro Fish ‘n Chips EQ. Dirt cheap and very effective.

 

My preamp has a big red boost button and a knob to control how much it boosts:

Fire-Eye Red-Eye

The pedal sounds better than any of the more knob- and switch-encumbered preamps I've used, and the boost is absolutely clean.

These guys talk too much, but if you fast-forward a few times, you'll get the idea:

 

Here's another approach using an EQ pedal. Set the preamp for your lead sound. Use the EQ to cut the volume and shape the tone for your rhythm sound. The HBE Detox EQ was made for this app but a GE-7 or something similar would work fine.