"Sorry but my map doesn't mark anything as Kern Basin. Trying to put your words into my map, does that mean to keep north of Ericsson Crags & Mt. Ericsson, keeping south along the mountains but not over them to pass those two water bodies (kidney bean shaped, and pac-man shaped), to get to Harrison Pass, which I'd assume won't be labeled but will be south of the next body of water? This I take it will take me to whatever trail goes east, running south of Gregory's Monument and onto Forester Pass?"
The Kern Basin refers to the drainage of the Kern River, south of the Kings Kern Divide. To get to Forester Pass from Lake Reflection descend the trail from Lake Reflection back to East Lake about 1/2 mile. There is an unmarked trail that leads east along a creek up to the small unnamed lake directly below and slightly NE of Ericsson Crags. Continue east up the valley between Mt Ericsson and Deerhorn Mountain and curl around to the south below Mt Stanford until you come to a broad, talus filled gully that leads up to Harrison Pass. There are a couple of strenuous moves at the top, but it is all class 2. If there is ice, it will be another matter entirely, but I suspect that will not be the case this year. Once over the pass descend about 1 mile south to Lake South America and follow the trail south 2.2 miles to the Upper Kern Cutoff Trail. Then turn left(east) and follow that trail .7 miles down to the JMT. Turn left and follow the JMT north to Forester Pass, 4.7 miles. I am using the notation and distances given on the Tom Harrison Mt Whitney High Country trail map. It is all the map you will need for this section of your trip. I hope this helps. Good luck.
"Is there a trail there leading to Forester or is it sort of a by sight thing?"
See above narrative.
"There's essentially a shown path on my JMT maps that goes through Thunder mtn., Mt. Jordan, Mt Geneva, Mt Ericsson, and onto Forrester pass. It seems perfect for my purposes but I was wondering if anyone has had experience with it."
There is no such trail to my knowledge, and I am very familiar with that area. Could you give me the name of the map?
"Thanks for the advice I'll totally be taking Harrison Pass if it'll let me to Forester rather than Lucy's or Milly's."
It will definitely get you there far more easily, and safely, that either Milly's or Lucy's.