The FEC was established as an independent regulatory body in 1975 and it already collects financial information, and investigates and audits campaigns. So collecting and vetting military records would hardly be a stretch. Actual investigations of claims like Blumenthal's and Kirk's would probably be exceedingly rare and mostly unnecessary, as the very fact of FEC oversight would discourage politicians from making such exaggerations in the first place. Knowing that someone was watching, and that lying about your war record was as quick and easy to expose as an illegal campaign contribution, you'd have to be reckless or insane to do it.
And what about those politicians who've been lying for years? Well, if they have ambitions for a political future, they'll have to 'fess up quickly. This will have the corollary benefit of getting any admissions out of the way early, so campaigns can be about actual issues. Maybe the scandal-hungry press will actually cover those issues. And maybe we'll be spared the unctuous sanctimony that disgraced politicians like Kirk and Blumenthal resort to when exposed--corralling the local VFW for a photo op, wrapping themselves in the flag, and whining that their military service is being "denigrated." It would be a better world for everybody.