While that double-standard is unfortunate I do not see how their ability to lie to someone should make that person tell lies. Is the alternate universe where the police are not allowed to lie to you is also one where if an officer makes a factually inaccurate statement during questioning gets the interview tossed from court? I suspect that it is and in that case I prefer this universe.
Aside from being insecure it is also not robust (fragile?). So, suppose Paul can lie to Alice and Bob, and they cannot lie to him. He can probably interview them separately, too. We can ask how the system will break, either inadvertently (fragile) or due to deliberate pressure (insecure).
Paul's later accusation (of lying) depends on his ability to remember with perfect accuracy all of his discussions with Alice and Bob; and if anything confuses him he might take it as grounds for the accusation. This leads to fragility: Paul's use of lies will likely create a situation where he himself is more likely to be misled, which he could then pin on Alice and Bob.
He could also put people in a stressful situation hoping to coax out the truth, only to coax out a lie. Paul can tell Alice that he thinks Bob is guilty, but that Bob has just put all of the blame on Alice and made her look awful. Paul says, "I'm going to have to put you in jail, Alice, if you can't give me something to incriminate Bob." She doesn't have anything to incriminate Bob so she lies; he might later discover the lie and bust her for it.
For that matter, Paul can stick to subject matters which Alice and Bob find embarrassing, and thus (advertently or inadvertently) stress them out into lying to him. Paul can suggest that even if his allegations are untrue, he will bring them to public light and their peers will judge them based on those allegations. (The example here would look like: Paul is pursuing a child porn ring which has nothing to do with Alice and Bob, but he thinks they're central to it. He tells them that if they don't immediately help him out, he's going to have to put them on public trial and then all of their neighbours and business associates will judge them for it. Then he starts asking questions about the other people in the ring, and they lie about it.)
This only gets compounded if he is allowed to lie to them on procedural matters. For example, Paul informs them "You're allowed to lie to me." Or, Paul informs them, "I'm allowed to imprison you if you say anything which I disagree with." It can be subtler and less coercive: suppose he says that their best option to protect themselves is to apologize for a related act to some appropriate third-party, Carol. So the think that they're lying to Carol (legally allowed) but Paul thinks they're lying to him (a federal offense).
And finally, even if Alice and Bob tell the absolute truth to Paul, it might be the case that Carol -- whether lying or honest does not matter -- contradicts their story and is otherwise trustworthy as a witness. Perhaps she really thinks she saw Bob that day, but in fact it wasn't him. Now both Alice and Bob have said that he was not there, but Paul interprets those as lies and prosecutes them for it.
I am more or less aware of these scenarios and they certainly raise some concerns about potential for abuse, but the problem is that the opposite scenario where federal agents or LEOs of any kind can not lie is not feasible.
I asked another commenter below for an example of someone being convicted of making false statements when they did not know they were speaking to a federal agent, I would like to see one as the lying in the example above was during a formal interview process.
I also find that a lot of these sorts of scenarios assume that the courts are incompetent, that the burden of proof does not still apply. The judiciary has a duty to apply the law in a reasonable way that complies with its intent - I doubt any judge would regard the intent of the false statements law to be to allow an FBI agent to tell someone to lie to them so they can then catch them lying and convict them. (This of course is why if you undermine the courts you undermine democracy.)
