In what case should the subject of a gerund phrase be?
You're right -- none of those sit right. They have heavy subjects, and English doesn't like that.
There are a lot of syntactic rules that fix this; the most common are Extraposition and Clefts, of one kind or another.
In this case, here are some sentences that sit better, all from the prescribed clauses, all synonymous:
- It shows his character that Bill took the money. (Extraposition)
- What shows Bill's character is that he took the money. (Wh-Cleft)
- It's Bill's taking the money that shows his character. (It-Cleft)