Thanks both for the heads up on that, will check it out. And yes she will be doing a dry run!
As I just had a couple of free hours and had a bit of shopping to do I just rode your course from Clapham Common to the end. I have a couple of heads up and a little tweak exiting Hyde Park for you which it makes it a better route IMO.
Heads up...
(i) You do need to be quite assertive making the right turn from Chelsea Bridge Rd into Ebury Bridge Rd.
(ii) There's building works along the whiole left hand length of Ebury Bridge Rd with a barrier meaning you can't get onto the pavement if you needed to. I imagine this won't be too much of a problem but I'd say your daughter needs to be confident enough to ride in primary along here if necessary.
(iii) The segregated cycle lane leading from Ebury Bridge Rd to Avery Farm Rd is easy to miss, particularly when there's a white van parked in front of it's entrance as there was today.
(iv) As suspected, there's no cycle route across Hyde Park. You can still enter the park (via a nice bit of filtered permeability heralded by the flags of the French and Kuwaiti embassies) along the route you posted but then you do indeed need to follow the path around the east side of the park until just after Speakers' Corner.
From there your daughter will be close to her destination but there's a little tweak which is better, and quicker/safer than the route you posted. Have a look at the image below. Your route had her going through or around the Marble Arch roundabout. Going around from Hyde Park would be unsafe and probably illegal because as soon as the trffic from the right is stopped by a red then the traffic from the left gets a green. Going through means being held up for 10 minutes by dint of having to wait for four or five light-controlled pedestrian crossings.
Instead follow the park cycle path around for another 100 metres and take this route....
Although Seymour Street is one-way for motorised traffic it's not for bicycles and it all felt very safe.
One thing I didn't do was cycle back via the same route as I was heading off somewhere else. I suspect that a few bits of the route can't simply be reversed but I wasn't really taking note.