Private equity’s largest lobbying group this week fired chief executive Doug Lowenstein, largely due to internal management issues. There also was a sense among board members that Lowenstein did not have adequate access to the halls of power during important policy debates like Dodd-Frank, despite his long family friendship with co-sponsor Barney Frank (D-MA).

The group has temporarily put its vice chairman in charge, while it searches for a replacement. To help move the process forward, I’d like to suggest that they consider Evan Bayh for the job.

Bayh is the former Indiana governor, U.S. Senator and perennial presidential tease. Since retiring, he also has become a senior adviser to private equity giant Apollo Global Management (APO) and a partner with law firm McGuire Woods. Both of those jobs are based in Washington, D.C., as is his regular pundit gig with Fox News. All of this despite Bayh’s 2010 claims that he was tired of politics and wanted to spend more time with his family in Indiana. In other words, he’s a lifer.

He also was remarkably uncontroversial as a national politician. Clearly a Democrat who publicly supported Hillary Clinton during the last presidential cycle, but also a self-described Blue Dog who was among his party’s more conservative members. Such a track record may not exactly give him street cred with either side of the aisle, but it also won’t cause doors to be slammed in his face. Such bi-partisan access is essential for a lobbyist in D.C., where the party in power often changes. Other ex-politicos currently working in private equity — Bill Frist, Ken Mehlman, Dan Quayle — can’t really say that (with the possible exception of Cerberus chairman John Snow, but he’d be a no-go due to the Chrysler fiasco).

Moreover, Bayh’s hiring by Apollo and McGuire Woods suggests that he still has plenty of friends in very high places.

Private equity currently needs someone familiar with the market — Lowenstein was a professional lobbyist who previously shilled for the electronics industry — and who has proven experience as a chief executive. Bayh has already spent seven months at Apollo, and earned his CEO stripes by running America’s 15th most-populous state. This isn’t to say that he yet knows private equity’s ins-and-outs like Henry Kravis or Steve Schwarzman, but he is a far better orator with a better grasp of regulatory process.

The counter-argument, of course, is that Bayh may not accept the job if offered. He pushed back hard against charges he was becoming a “lobbyist” for Apollo, with a source insisting that Bayh would neither register as a lobbyist nor advocate for its interests with elected officials. But that always has sounded like a technicality. Apollo doesn’t need Bayh to tell it about the regulatory environment. It can just pick up a newspaper, or outsource the job to the Private Equity Growth Capital Council.

If Bayh isn’t pushing Apollo’s case — in a perfectly legal and appropriate manner, I’m sure — then he isn’t earning his paycheck. This would simply be doing a similar job on a grander scale. Moreover, he’s already taken jobs at odds with what he said upon announcing that he wouldn’t run for a third term in the U.S. Senate:

“At this time, I simply believe I can best contribute to society in another way: creating jobs by helping grow a business, helping guide an institution of higher learning or helping run a worthy charitable endeavor.”

Pretty sure he’s not doing either of the second two, either with Apollo, McGuire Woods or his talking heads gig on Fox News. So let’s stop with the pretense, and get on with it.

Private equity needs a new voice in D.C., particularly if Mitt Romney becomes the GOP nominee and the industry comes under additional public scrutiny. Evan Bayh should be the man to give it to them.