Hi:
Regarding entry into ARG you are correct, Visa Exemptions: Nationals of Thailand for a maximum stay of 90 days. Nationals of Canada with a normal passport for a maximum stay of 90 days.
The steps for getting married become clearer as you go. The administrative process takes time. Initial part can be done online, the rest is in person. You also need a certificate of where you are domiciled (can be a hotel), which you can get as a tourist. I believe the blood tests are no longer required. When we did it they were required and that is a separate procedure done in the private or public clinic. Expect long waiting times. The day before the ceremony you must go back in person to present photocopies of all documents as well as originals of your IDs/passports and the IDs of the 2 witnesses (who must be legal residents or citizens of ARG). There is a procedure to be followed for interpreters. The whole process is very bureaucratic and time consuming. I do not know if there is someone who can handle all this for you, perhaps someone may contribute from this group. Lack of Spanish makes the whole process more frustrating in a country where administration is already burdened by inefficiency.
One thing to keep in mind: if you need to get a copy of your marriage certificate in the future for use abroad (and you are located outside ARG for example) it is a procedure you must complete online and then wait around 2 months for it to arrive at the embassy where you are (if it arrives at all due to it being lost in the system). The whole process is time consuming to say the least and burdensome to an ordinary citizen. The certificate is in Spanish and you will need to translate it for use elsewhere (after obtaining the correct stamps, including the Apostle of the Hague). It is possible to give someone a power of attorney to do this for you, but you would be creating additional work for you which requires extra time to execute and in the end may be a headache.
Out of curiosity, why BsAs? I am asking because the two of you as non residents can do this much easier in Copenhagen for example. The certificates are done in English and it is super simple. The ceremony is in English. A Thai national who is in de facto relationship with you should have no problems getting Schengen visa to accompany you to the EU. And if in the future you have to get a copy of your certificate, getting one from Denmark is a piece of cake compared to Argentina.
Unless you have very specific reasons to do it in ARG, I would not bother with it.
Let me know if I can help you further.
Appreciate all the effort put into the reply. I'll definitely look into Copenhagen as well, doesn't seem bad as long as we can preform the same-sex marriage legally and it's recognized in Canada that is an option for sure. We're still early on in our search and have 3-4 months before we're going to start crunching the numbers and booking flights so anything is game!