While in 7 days, you could
make about half the track (pushing it a bit I feel), the trouble is access points at about the halfway mark.
From our walk perspective heading east, after Ellery, you have to get to Standley generally, if using the standard transport company (Larapinta Transfers).
Some of the best sections I can recall was up through Hugh Gorge and the following ranges.
If I was only going to do one bit, that would be it (Ellery to Simpsons).
That would fit roughly within your timeline and suggested experience etc, and not be pushing too hard, time to soak in the surrounds a bit.
Actually, out of interest, I just looked up the Larapinta site for confirmation of some of our stops http://www.larapintatrail.com.au/multiday.html
(then I look behind me, and there's the Lara strip map on the wall !!
This web page has the section highlighted as one of the best sections.
We did the whole thing from the western end of Mt Zeil Range (off track to Redbank, then the LT) to AS in 19 days, with a food drop somewhere in the middle, must have been at Ellery.
We just had to bag Mt Zeil when out there, Glen Helen dropped us out there by 4WD.
We had one rest day at Jay Ck, after 3 or 4 days of continual rain, and it was still bucketing down all that day, was so nice sitting in the shelter playing cards and resting up . . . the creeks through Hugh Gorge we were wading knee deep, right through to Simpsons Gap, when it finally went further east.
It was as cold as I can ever remember Tassie being !! Only saw 3 or 4 other walkers the whole trip.
Will never forget it, was one of the best walks I've ever done with my mates