I'm trying to play a Burzum song and the tremolo picking starts at the A string and you must go to the B string and back down quite quickly (4 picks per string). The problem is that I tend to rest my palm softly on the lower strings on the bridge (if they're not required) but when I go from the lower strings to the higher my palm slightly rubs them as it moves down the bridge making an undesired rubbing sound.

What's the trick here to do this right? Is your picking hand not meant to touch the bridge/unused strings at all, sort of hover over them while picking?

Here's the riff I mean if it helps. Each note is struck 4 times before moving to the next.
It should be muting the strings below, i.e.

When you pick the A, you mute the E, when you pick the D, you mute the E and the a strings. Is that what you mean?
Not really. I don't have a problem with hitting strings I don't want to hit but rather while doing that muting of strings I'm not using I will move my hand down/up the bridge and it makes a rubbing sound sort of.

See the riff, it goes up and down, up and down. If the strings in another hypothetical riff are confined to two strings next to each other (eg. E and A) then I don't have a problem at all.

So basically I want to know what I should be doing with my palm when tremolo picking:
rest it on the bridge but just try to avoid that rubbing somehow OR hover my hand above the strings/bridge and only touch the strings with the pick alone.