Both of those bands use MUCH higher gain amps. Tom DeLonge is known for using Mesa Rectifiers on Blink's classic albums, and sum 41 used to use Kranks - which are no longer in business, but we're done of the highest gain amps imaginable.
If you put any sort of humbucker in your strat, even a used or cheap GFS pickup you can find on reverb, that will help a lot. I think you can get close enough with your amp and DOD pedal.
I would try engaging the "more drive" button, but keeping both your preamp and drive levels relatively low. Then, use only enough bass to not sound thin, and turn your mids up more than you normally would. The mids in this amp are really just in a different place than the amps you're trying to emulate, but you still need em. For rhythm sounds, I would keep your treble on the lower side, and use the presence knob sparingly. For lead tones, or in a single guitar player setting, I would almost dime both, depending on your pickups.
This amp setup should be an aggressive clean sound, if that makes sense. Your overdrive pedal should push the amp right into the territory your looking for. I would keep the tone knob close to noon, then start with the drive on 2 o'clock, and turn the level control up until your tone is driven, but not fully saturated. If the sound gets woofy, cut your amps bass further, and/or turn up the tone on your pedal.
Keep in mind, those album mixes have 2 to 4 guitar tracks, which is what provide that dense but clear guitar sound. This setup should get you a good live pop punk sound.
P.s. while waiting on your pickup swap, if you lean into the treble and reverb, you should be able to get a decent Dead Kenedies tone - totally different, but equally fun.