To make a LVM backup of a MySQL database, would I have to reduce the logical volume size?
Unfortunately, yes, you'll need to reduce the size of your root LV.

(You could reduce the size of swap, too, but that's already a fairly small volume)

Even if you can use that 4MB, a snapshot of your root volume will become invalid quite quickly when LVM runs out of "free" disk space.

You get prompted with a text box asking how much of your disk to use when you do an Ubuntu install (even if you select "use entire disk"). If you're building systems to muck around on, or virtual machines, I recommend setting this to 90% (leaving 10% for LVM) - you probably won't miss the space, and you never know when it'll come in handy.

The mechanics of how to shrink the root volume aren't too hard to find on the web, but questions about that are probably best suited to unix.se.

Update: Make sure you don't rely on LVM as your only backup strategy - if you change enough data on disk and your snapshot runs out of free LVM space to use, the snapshot will simply stop working. Using LVM is one of the only ways you can take file-based backups of MySQL (esp. in conjunction with FLUSH TABLES WITH READ LOCK) without shutting down the server, though, which is quite handy.