Of course, do whatever you want, but after 4 months doing 12,000 K (7,000 miles approx) and five South American countries here is a suggestion...It will take you a full day air travel with delays/recovery time to fly into Buenos Aires, you can rent motos from Mariano at MOTOCARE, I believe he is the only agency that rents to indivduals, without having to go on a group tour with a guide. Absolutely make reservations in advance, so motos will be waiting with paperwork/insurance etc. Crossing borders, country to country is possible with rental bikes, but requires special paperwork, not always recognized by all officials, and could cause extended delays at borders. Don't foget to plan a full day for return travel to USA. That leaves you 12 days. I suggest two nights in Buenos Aires and ten days riding Argentina.
Take it easy in Argentina, plan on 150-200 kilometers a day (check a good SA map and don't forget distances may be in kilometers, not miles)especially if you are doing gravel, dirt, mountains etc. Camp or allocate time to find inexpensive hotels that have secure parking for motos. Sometimes finding a place to sleep with motos parking takes a couple hours. Keep in mind summer here is December/January/February. It is cold now in southern Argentina, and Buenos Aires is in for some rain (Las Santa Rosas) "storms" soon. Otherwise northern Argentina is good.
Check your weather, and plan your ride within Argentina. You can find everything here that you would find crossing borders, with a lot less hassel.
Let me know if you want more details about moto renting and I will give you Mariano's phone number, he speaks excellent English.
I am an expat living in Buenos Aires, and will help all I can. xfiltrate.