From what I have read, Bundy did horribly in law school so any future as an attorney is out the window.
Having lived in SLC, with his background in politics, being a staunch Republican, and a member of the LDS Church, he definitely would have moved up both the political and religious ladders (something that is required for both). His psychopathology would have served him well in office. Charisma goes a long way and for the rest, staff can conceal the negatives and accentuate the positives. He EASILY could have made it into the House. Senator is likely pushing it too far.
His desire to murder and rape could likely be curtailed by busying himself with campaigning and civic responsibility. The man was a narcissist who needed power in some form. His murders only increased the worse his life became and only a chance discovery by a highway patrolman led to his arrest. Make him a bishop or higher in the Church, keep increasing his political clout, and marry him into a publicly required domestic life and the murders likely decrease in number (and his ability to travel extensively due to being a Congressman may further conceal the connections of his crimes).
He still likely gets caught by the mid 80s or early 90s if he doesn't fully redirect his aggressive nature from murder to politics, something BTK was able to do for some time with his employment. If he makes it to his forties, Ted's need to rape and murder likely drop (an odd discovery psychologists have made about serial murderers).