I've done the JMT and over half of the SHR.
I like the idea of doing a creative combination of the JMT and the SHR. In fact, if you have plenty of time, I would add some layover days/side trips to your itinerary. People hike in for days JUST to see places like Darwin Basin, Sixty Lakes Basin, Bench Lake. The JMT routes right past those places, so why not get off the trail to explore them while you're there?
This way you can see some things that are new and exciting for you; without overwhelming your hiking partner in case she finds she prefers the JMT experience to the more hard-core off-trail. You can "bail" from the SHR or from a side-trip back to the JMT at any time.
Something like the following could be extraordinary,
Happy Isles to Thousand Island Lake via JMT. Possible short side trips to Half Dome, Lyell Glacier, Marie or Davis Lakes basins. Lyell Glacier is expected to disappear by end of century, it is right off the JMT and we are idiots not to stop and see it (I include myself in this).
At TI Lake, leave the JMT. Walk along the northern side of the lake to its far southwest side, and there, hook up with the SHR southbound. Much as you loved the JMT from TI to Garnet Lake, the SHR through here is much more spectacular IMO, with fewer people! Continue on SHR till Minaret Lake. At Minaret Lake, bail out via the Minaret Lakes trail back to Devils Postpile.
Take the shuttle back to Mammoth and then take the Lakes Basin shuttle to Lake George. From the Lake George trailhead, take the official trail that gains the Mammoth Crest and there, rejoin the SHR. (By starting from Lake George, you are avoiding the very WORST section of both the JMT & the SHR: The hot slog through dead forest immediately southbound out of Reds.) Follow the SHR to below Duck Lake, where the SHR and JMT become one southbound.
At Tully Hole, you'll have to decide between taking the JMT or the SHR. Both are gorgeous, the Silver Divide never disappoints. Either way, try to build in a layover day, if on the SHR you could spend it exploring Grinnel Lake, and if on the JMT spend it exploring some of the lakes near Silver Pass. If you are inclined to show you friend the VVR, which is what I would do, the JMT will get you there faster.
South from Mono Creek or the VVR, I agree with someone here that I hate that ascent up Bear Ridge on the JMT, but I also haven't done this section of the SHR yet so I have no beta for you. Heard the SHR is both gorgeous and high/challenging through there. If you decide on SHR, you might want to walk straight through from Mammoth to Piute Pass and resupply in Bishop. If you decide on the JMT, maybe a layover day to explore the lake basins east and northeast of Marie Lakes.
From Piute Creek to Leconte Canyon the JMT and SHR are mostly one. IMO you have 3 choices: JMT; SHR over Snow Tongue Pass; or SHR variant over Alpine Col instead of Snow Tongue Pass. See High Sierra Topix forum for more beta on why most sane people avoid Snow Tongue. Snow Tongue & Alpine Col are quite far east from the JMT, so which route you choose might depend on where you most recently resupplied: VVR/MTR to the west or Piute Pass/Bishop to the east.
Either way, take some time for a layover-day side trip from Evolution Basin: Darwin Bench, Davis Lakes, or Ionian Basin.
Once into Leconte, definitely take the SHR from Dusy Basin to Palisade Lakes! - unless your friend has freaked out at the x-country. Really amazing through there. SHR and JMT rejoin southbound from Palisade Lakes.
After going over Mather Pass, you can
1. End your trip by going south(west)bound on the SHR and finishing at Roads End.
2. Continue southbound on JMT over Glen Pass and from there, bail east over Kearsarge or west to Roads End. Be sure to take layover days to see Bench Lake, Sixty Lakes Basin and Dragon Lake.
3. Continue on JMT to end at Whitney. More possible side trips down there between Forester and Whitney, including Wrights Lakes Basin, Wallace Lake, Arctic Lake.
How much time do you have? What I find is that people really rush their trip to make it to Whitney, as if Whitney is the Holy Grail. And the result is that they spend every second on the JMT dust highway and miss ALL the truly spectacular side trips along the way that are just a mile or two off the JMT. Unless your friend is dead-set on Whitney, consider choosing one of the slower-moving options ending at Roads End or Kearsarge/Onion Valley. Another way to save time is to skip the Happy Isles - Tuolumne section of the JMT, as this is kind of a boring section of the JMT if she is not dead-set on doing Half Dome.