I watched a video lesson on that song, is there a muted bend? I saw a lot of very quick slides. I guess if I did need to do something like that, probably just muting with the left hand would be possible. Just put your fingers down against the strings quickly like your grabbing the neck. But use palm muting when I need to mute. There are probably videos on this site are certainly YouTube that will show you how to do that.
I’m not sure what kind of a guitar you are playing, but if you’re playing an acoustic guitar, non-amplified, I could understand how you might still be getting that thump, thump thump as you pick over the muted string. If you’re playing through an electric guitar, turn the volume up.
One more pizza for the road!