I have the same devices you do (SGT 10.1 and Galaxy Nexus), and tether then all the time. Rooting the SGN is the way to go. I've had mixed results using various add-on tethering apps on my OG Droid with Froyo and Gingerbread, but now that tethering is built right into ICS it's a whole new story. Many of the custom ICS ROMs enable use of the ICS tethering without signing up for the extra-cost VZW service, so it's very easy to use and you get very fast and sold connections.
FWIW, I'm personally running the AOKP (Android Open Kang Project) ROM. Besides lots of other benefits, it has a tethering toggle along with the other power control toggles right at the top of the notification pull-down. So it couldn't be much easier: one click and the hotspot goes live; the tablet instantly sees the already-remembered SSID appear and connects automatically. I'm seeing full 4G speeds on the tablet (usually about 10-15 mbps) - unlike the pre-ICS apps where I generally saw lower speeds on the tethered device than I did on the phone itself. Be aware, though, that 4G tethering really chews up the phone battery, so you'll want to have the charger connected if you plan to tether for any length of time.
So go for the rooting and a custom ROM - problem solved!