No, Argentina will not reimburse your reciprocity fee
Nor should you give it a try and see if it works.
Reciprocity fees (and visas, for that matter) bring in substantial and much needed revenue to countries, which they then reinvest it in their economy, in the way of improved touristic facilities, better immigration services to travelers, newer airports, roads, national parks, etc.
Contrary to what most people think and feel, it is not a penalty to visitors, nor should it be interpreted that way. At least not when it comes to developing countries. I hate to sound all Sarah McLachlan here, but for the cost of $160 over the course of 10 years you’re helping maintain infrastructure and create hundreds, no thousands of jobs for a developing economy.
To put it into perspective, the US is analyzing to include Argentina in its visa waiver program, but make no mistake, the reasons for doing so are not so much a result of warming relations between the countries as it is an economic one. A study published in 2014 found that when countries are included in the visa waiver program, the United States experiences a substantial increase in the number of visitors from that country, 16.4% in the first 5 years to be exact. For example, if we included Brazil, Hong Kong, Turkey, Israel, South Africa and Poland, it would mean an increase of 600,000 visitors, which would bring in an additional $7.66 billion in tourist spending in the first 5 years and create at least 50,000 more American jobs in the tourism sector.
The only valid reasons to claim your money back are for a Duplicated Debit (the system charged you twice), Fee Already Paid (you already had paid for 10 years previously and oops you did it again) and Argentinian Citizen (you were charged when you shouldn’t have been in the first place).
It is easy to lose perspective of how important these things are in our zeal to reduce the cost of our travel, but may I suggest that if cutting costs is a priority, we do so by going from a category 7 to a category 6 hotel, or maybe skipping that brunch at a Michelin rated restaurant? Just because we’re in the points and miles game it does not mean we cannot be responsible socially aware travelers. The world will thank us.