Seriously. For years I've played with an incredibly hard attack (fingers) and I've learned to slow down, bend that hand around a bit and just use the thumb, just like The Sharp One above. Makes all the difference, and you can mute with the rest of your hand (or not) as you see fit too. Lots of interesting possibilities.
I don't hear anything in that part that would require a pedal. It's more about learning to play legato cleanly.
Aside from that, some things that can help smooth out your tone for that type of part if playing pickstyle are favoring the neck pickup (if a two-pickup guitar or bass), rolling off the tone knob (or reducing treble generally), using a fatter pick, and using one of the pick's rounded ends instead of the point part.
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