Cat I'm not sure I share this interpretation:
The implication in Caroline Lucas' tweet is that any investigation underway at present - and we've been assured an investigation was begun immediately - wasn't independent.
I read it as no investigation has yet been instigated as far as Lucas was concerned and was aware and she believes it needs to be independent but there hasn't been any formal discussion of how such an investigation should be carried out yet.
She certainly, however, doesn't seem to be on the same page as Aimee's tweets which are about being framed as guilty purely by association by family ties, which don't take into consideration Aimee's independent actions.
In other words, Caroline seems to be publicly saying the contrary; an investigation of both David's role AND Aimee's independent actions are BOTH legitimate and need to be examined in their own rights for lessons.
Aimee stated that she had the full support of the party at this time (perhaps a bit like a football manager on the edge) and that tweet seemed to imply thats on both on a political as well as personal.
The reality might well be different. Caroline seems to give that impression.
You can't have an independent investigation into a member's actions when a decision has already been made that Aimee has a political future with the greens. If the independent investigation finds that Aimee's conduct isn't acceptable, then how can the party continue to stand by Aimee politically (it can't).
Caroline suggesting that an investigation needs to be independent does remove the transphobic lobbying power to a large extent. Its an attempt to neutralise things a bit. How can anyone object to an independent investigation? (they will btw).
But it also begs the question, how can Aimee not be formerly suspended in the meantime? Stepping back from duties and continuing to use the power of the party (which is inherent in using a twitter handle which contains the words green party), are also not sufficient.
It merely places the party in a position where they'll be in a right pickle if an independent investigation finds Aimee has made actions which are highly questionable.
I can well see this imploding into spectacular green party infighting over what to do yet.