You could consider the term scapegoater.
While this exact word does not appear to be listed in the stock online dicitonaries, it does appear in literature, beginning around 1910, and increasing in usage after 1940, as evidenced by this ngram.
The meaning of the much more used term scapegoat is
a person who is blamed for the wrongdoings, mistakes, or faults of others, especially for reasons of expediency.
The verb form is defined as
make a scapegoat of:
secret police scapegoated a few of the ringleaders to put an end to the issue
Jack's actions could be characterized as scapegoating
They also could be considered projection, a psychological concept defined as
the attribution of one's own ideas, feelings, or attitudes to other people or to objects; especially : the externalization of blame, guilt, or responsibility as a defense against anxiety
It also sounds like Jack considers Jill his nemesis
a long-standing rival; an arch-enemy: