Lance Armstrong... greatest doper there was or ever will be

But why on earth would they risk themselves by bribing people, or being "in on the secret"?

To answer the second part first, they would have wanted to position themselves with "plausible deniability", which oddly really requires a knowledge of what you're going to be exposed to and how you're going to make a plausible case that you didn't know. Obviously it helps if you can place several layers of separation between yourself and the crooks, so that you can let the ambiguity build up as contractual obligations are passed down, and assurances are passed up, the chain, but you also want to bypass all that informally, preferably secretly but certainly orally rather than in writing, and get the truth direct from Lance to senior management at Nike.

On bribery, it needs a lawyer who has read the full FCPA to decide whether Verbruggen is a 'foreign official' within the meaning of the act, but on my preliminary scan of the matter it is quite likely that the UCI doesn't fall within the scope of the law (and Nike have expensive lawyers, so I'd expect them to have checked). If this is true, Nike have no legal exposure as a result of bribing Henricus, just exposure to reputational damage; they have shown over and over to be good judges of when a little investment ($500k to HV, questionable employment practices at their manufacturing subcontractors) yields such vast returns that a bit of whining from bleeding heart liberals can be laughed off.