No Mute Tool in LE? What are my other options?

One way is to click the track you want to edit, selecting it.
Then drag the cursor across the timeline at the top over the space you want to delete. That space will turn gray so you can see it's selected.
Hit the computer keyboard's Delete key.
That won't mute it, but you won't hear it anymore, either.
You can also use the Split tool to cut out, then delete that part.
Or you can right-click the clip, select Envelopes>Volume, add nodes on the line that appears and drag the volume of that part all the way down to -INF.