I would like to clarify somethings here.
One is that the German flirting with Argentina was because the Germans saw us as the most Aryan people in South America.
Nevertheless, Argentina joining the Axis is really difficult. For two reasons. One is that majority of the common people(except some descendants of Italians who were pro fascist) was against the Axis, and would have caused more problems in a time they were already causing many.
The other and far more important, is that taking a side is not good for our economy in the short run. Being neutral you can trade everybody. If you can with the USA and UK excellent. If Germany for some reason manages to take the UK or to finish their naval supremacy, then we will trade with Germany.
Argentina joined the Allies after pressure that the trade will be restricted. This pressures didn't came earlier because the British needed our beef.
Argentina joining the Axis depends in a almost complete victory of Germany and Japan, which is really implausible.
It's certainly more probable to have Argentina join the Allies earlier as Brazil did and gain some economic aid as Brazil did. It would require the pro-Axis militars not being in power(there were many pro-Allies militars) and then joining the allies.