What advantages are there to operate a corporate flight department under Part 135 as a separate entity instead of in-house as a Part 91? This question specifically involves FAA regulations in the U.S.
Some U.S. corporate flight departments will set up a separate business entity for their flight operations to limit the liability of the parent company. Because this business operates with on-demand passenger transport for compensation as its primary business, it must be operated as a part 135 (or 125 if they are running big ships) instead of part 91. In some cases, this new subsidiary will provide additional revenue stream by servicing outside clientele in addition to meeting the travel needs of its parent company, and sometimes they remain exclusive.
If a corporate flight department doesn’t want to use their aircraft to supplement income with charter flights, is liability the only reason to set up and operate as a Part 135 instead of keeping it Part 91?
Are there any other reasons, aside from liability and supplemental income, to bother with a 135?
There is no reason you can't operate as a part 91 department but simply adopt the standards of 135.  Government contracts work this way.  It's all pt91 work but adheres to pt135 rules.
Actually holding a 135 is a lot of work and is expensive.  The training and checking requirements are onerous for your average flight department.  How do you feel about potential career limiting checkrides every six months?  You also need to keep the planes up to a higher standard.  This means observing TBO hours and calendar times.
You don't mention aircraft types, but Continental requires piston engine overhauls at 12 year intervals or upon reaching the time limit, whichever is first.  Even a piston single becomes expensive to operate.