How can one deal with a half-brother who consistently lies, watches pornography, harms his siblings, and raises his voice at his father, while being keen on having deeds accepted on Mondays and Thursdays?

If a brother lies, mistreats his parents, and watches pornography, he is in danger, and he must be advised, enjoined to do good, and forbidden from evil.

As for shunning him due to his sins: The determining factor is benefit. If shunning him helps in his reform, then it is preferable. If maintaining ties with advice is better, then maintaining ties is preferable.

Your boycotting him due to his sin does not fall under the warning of the hadith: "Deeds are presented every Thursday and Monday...", because this warning is for those whose shunning is for worldly matters.

A believer should be merciful to sinners, hoping for their guidance and repentance, and disliking the sin, not the sinner himself.