Taking it as a given for a moment that a truce or peace could never be established and all-out war to the death is the only option, defeating the bug comes down to understanding where the bug is strong and weak.
You could...
Shoot the Bug - A simple solution, and one taking place all along the front lines.
Advantages: reduces the volume of bugs in an area.
Disadvantages: the bugs breed too fast and too greatly outnumber humanity for this to be a permanent military solution.
Gas the Bug - Using chemical compounds and neurotoxins lethal to the Arachnid life forms, humanity can attempt to poison all bugs it encounters.
Advantages: easy deployment from orbit or by fast moving MI gas deployment teams minimizes the need for vulnerable ground forces.
Disadvantages: bug technology is based on genetic engineering. As in terran insects, immunities will almost surely be developed, making this a temporary situation only.
Confuse the Bug - Using humanities developing psychic ability, Intel develops a method in which the connection between the Brain bug and its subordinates can be severed.
Advantages: no need to exterminate the bug, because it no longer can coordinate its forces into an interplanetary threat.
Disadvantages: not yet demonstrated as possible.
And so on... clearly the weak point in the empire is the connection between brain and bug. Break that, and the rest is cleanup. Already, we know human psychics can (in the movie) tell what a Brain is thinking. In the cartoon this game is closely based on, we see that a psychic can literally explode a Brain bug, and that the bugs can fully assimilate a human being. It could be imagined that a powerful enough psychic would themself act as a brain bug and command Arachnid forces, using their own numbers against them.
But, if Barcalow's fate as a hybrid in the cartoon is something that could happen to anyone, it could mean that humanity will not go extinct at the end of the bug war regardless, because they will have become a part of the Arachnid species.
You could...
Shoot the Bug - A simple solution, and one taking place all along the front lines.
Advantages: reduces the volume of bugs in an area.
Disadvantages: the bugs breed too fast and too greatly outnumber humanity for this to be a permanent military solution.
Gas the Bug - Using chemical compounds and neurotoxins lethal to the Arachnid life forms, humanity can attempt to poison all bugs it encounters.
Advantages: easy deployment from orbit or by fast moving MI gas deployment teams minimizes the need for vulnerable ground forces.
Disadvantages: bug technology is based on genetic engineering. As in terran insects, immunities will almost surely be developed, making this a temporary situation only.
Confuse the Bug - Using humanities developing psychic ability, Intel develops a method in which the connection between the Brain bug and its subordinates can be severed.
Advantages: no need to exterminate the bug, because it no longer can coordinate its forces into an interplanetary threat.
Disadvantages: not yet demonstrated as possible.
And so on... clearly the weak point in the empire is the connection between brain and bug. Break that, and the rest is cleanup. Already, we know human psychics can (in the movie) tell what a Brain is thinking. In the cartoon this game is closely based on, we see that a psychic can literally explode a Brain bug, and that the bugs can fully assimilate a human being. It could be imagined that a powerful enough psychic would themself act as a brain bug and command Arachnid forces, using their own numbers against them.
But, if Barcalow's fate as a hybrid in the cartoon is something that could happen to anyone, it could mean that humanity will not go extinct at the end of the bug war regardless, because they will have become a part of the Arachnid species.