Ive only paddled the river below the lake once, in June.  I can see where it could get low in mid-summer and cause some issues.  the first obstacle is an old bridge you have to work the canoe under with a fair amount of debirs beneath it as I recall - lower water woule likely necessitat a portage there.  if I recall it correctly, the next portage indicated on the map is really just a low water level portage around a rocky area - we just paddled over the rocks. 

Another consideration, while there isn't very much current in the river in most places, low water would likely mean even less current, adding to the time it would take you.  I seem to recall about a 14 hour day from the island campsite in Cache to Kawnipi L, and that was with likely "normal" water levels for end of  may early June - the river crossing on the way to Cache L was about a 100 yard paddle when we did it, so water level was still spring flood there - in summer, that river crossing is maybe only 6 or 10 feet wide.   If you can't do the whole river in one day, you'd better make at least 3/4s of the way or so, as there are no campsites along the way - lower down, you at least have dry ground (in the area of the lower few rapids) - upper is a lot of alder swamp and the trees are a long ways from the river

I would guess that you would have problems in a low water year by mid-summer.   take 2006 for example, when the water was very low in June before the rains came, would have been a bad time to try the upper river.