I just went through this entire experience.
I had been living here for about 5 years +/- on tourist runs + actually leaving for work/family reasons every 3 months. When we got married and changed our plans to stay here I started the process of getting everything done.
Our lawyer (family friend) agreed to assist with my case and advised me that I didn't need the FBI check since I had been here for longer than 3 years. I had never been out of the country for longer than 1 year, and I had been in Argentina for at least 3 years (7 since my first visit in 20008 and I have never been away from Argentina for longer than 4 months in any one stretch)
So everything was presented this way and accepted. I got my Precaria and then waiting for everything to process. I renewed my precaria once without any problem, but then the second time it wouldn't renew online and I had to go down to migraciones. There they still never told really told me what was the problem. I had to have my lawyer look at it and my case had been flagged as "falta de documentacion". She submitted an official note saying the requirement did not apply to me, blah, blah, blah. After another 3 months, I went to renew and they refused. They said I was missing documents, blah, blah blah. After they got quiet militant with me I finally found someone who explained what was the problem.
The law says that you must have been residing in Argentina for at least 3 years and not returned to your country of origin for "longer than a period of 1 year" in order to forgo the FBI check. But they decided to apply ALL of my trips to the US over the last 3 years and combined they said they equaled just over 13 months and thus I needed an FBI check. Ironically enough, we went through my passport and all my flight records for the last three years and realized I had actually spent roughly 320 days out of the last 3 years in the US. I tried presenting this information to the people at migraciones (migraines as I call it) and they just didn't care. Finally, after literally talking to 3 supervisors and 6 other low-level peons, one finally just leaned in and said to my (Argentine) wife "look, you are from here...you know how it works. Explain it to him....it doesn't matter what the law says. The director says he needs his American background check...even if you are right, it doesn't matter, the director wants it and thats that." Even my lawyer went and presented notes saying this did not apply in my case, but they didn't care (note, my lawyer has successfully argued cases before the supreme court, and handles immigration items regularly...so she is perfectly capable)
In the end, I had to beg them to give me 30 day extension of my precaria so I could go to the states and get my FBI check. I left a few days later to go get it in person. One I came back I showed them a print out of my report (in english) and said I needed more time for the official version to arrive. No questions asked they gave me 3 more months on my precaria (irritating because before I left they swore it was impossible to have a precaria extension for longer than 30 days)
MY advice would be to just get the check if possible. Otherwise, even if the law is on your side no neat migraciones seems to care. If you are even lucky enough to get them between mate breaks, they won't give you any consideration in the matter, especially if it requires them to think even slightly outside the normal thought processes they have each day.