Under normal circumstances, the best vehicle to investigate this matter would be a special counsel. I recognize that recent history makes this proposal controversial. But regardless of your opinion of former special counsel Robert Mueller, he conducted a professional, thorough investigation that resulted in a comprehensive report that extensively documented his findings. There are other well-qualified investigators available to serve in this capacity.
The real problem with naming a special counsel is not recent history but the politics of the Department of Justice. Given the allegations against Barr, a special counsel makes perfect sense here to avoid the appearance of impropriety. But it’s hard to imagine Barr (or the deputy attorney general, in his stead) appointing a special counsel. Given that reality, it will be up to Congress to investigate this matter itself, either pursuant to its oversight authority or as part of another impeachment inquiry.
Congress can’t investigate as well as a professional criminal investigator could, but they made the most of the tools they had during the recent impeachment inquiry. There is no reason why they couldn’t continue their work.