I did the reverse route (WP to HSM via New Army) a couple years ago in July. I camped just above Guitar Lake and then at this really nice campsite on New Army trail a bit north of the Siberian Pass Trail. Soldier Lake had a lot of mosquitoes, so I just kept walking a bit further until I found this site.
I had to do the whole thing in 2.5 days, so what you are planning is doable but it will be a bit rushed. I wanted to do Langley the last day, but I had to be picked up at noon at HSM, so no time for that. I definitely could've done it if I had the full three days.
Hope this helps