I watched the Senate Permanent Subcommittee on Investigations when Duelfer testified. Lieberman had actually read Duelfer's report and was highly appreciative of Duelfer's efforts, as opposed to Levin, who kept trying to equate the US role in the UNSC as equivalent to France and China's wholesale theft of OFF dollars.
No, I don't like Lieberman's performance in the Gore campaign (who could?), but, as opposed to social issues, Lieberman has never wavered on security issues since 9/11.
I believe he could do the job, depending on whether he can be trusted to hold loyalty to country above partisanship.
Your trust would be abused by Lieberman at DHS. See #75.