If this was indeed an investigation passing usual legal requirements with the intent to build a case against her, I'd imagine that any responsible investigator would at the very least choose to do that as well. If they repeatedly searched her at the border (thus essentially abusing an avenue which had its legal hurdles significantly reduced for the ostensible sake of national security and being able to deal with external threats that the internal legal system had no time and opportunity to handle through regular channels) but did not choose to obtain a warrant or otherwise start a legal process that comes with the normal set of safeguards, I'd say that strongly seems to indicate they did not have enough of a case against her - or, in other words, that an investigation was perpetuated which according to traditional legal standards should have been discontinued.
Now, of course you may argue for the traditional standards/safeguards themselves to be weakened (e.g. by saying that while searches of one's home should remain subject to the same legal restrictions, certain parts of the government now also shall have the privilege to conduct some sort of "preliminary searches" which do not come with the usual safeguards to arbitary ends which might include a more formal investigation being started, at their full discretion), but this is not how I usually see this debate being framed.
I also have no idea if the FBI pursued it further and got a warrant. They're under no obligation to inform someone that they're under investigation. They may very well have started up a case against her that's still open, or may have investigated her and eventually dropped the case, or may not have done anything at all. In any case, they've never charged her with a crime and she's never been arrested.
It definitely wouldn't be a matter for the military police - since she isn't a member of the military she isn't subject to the UCMJ. The only reason they would have done anything in Iraq would be because Poitras was in a war zone outside US territory. They, again, would have had to hand her over to the DOJ for prosecution.
On the other hand, telling the investigators that she wasn't the person filming at the scene of the attack and then later recanting it is certainly enough to raise some eyebrows. Certainly not enough to convict someone of a crime, though, which is likely the reason why she was repeatedly searched and questioned but never arrested or indicted. She might call it harassment, but the authorities call it investigation.