> The room reservations via the Apimondia site don’t have Sunday evening available.  

Then call the hotel directly (many in Chile use "WhatsApp" VOIP), or for the big US-based chains, call their reservation center.

I've helped to organize small conferences, so I know the artifact of the "arrangement" at fault here.

Here's the deal - conferences get "good rates" (better than "rack rate") because they guarantee to pay for a certain number of rooms.
The hotels love this, but some only allocate the number of rooms guaranteed by the event planner.
The event planner puts these limits into their "official" website, and when they all sell, it looks to the uninitiated like the hotel is full - NONSENSE.

Pro-Tip: no hotel is really ever completely full, you just need to place a few more US dollars or Euros on the counter as an advance gratuity.
But can you get the "Apimonda rate" from the hotel if you book directly?  Very likely yes, as this is no more than the usual "Corporate Discount" it is NOT the best rate possible, as you'd well know if you were a member of the Delta Million-Mile club, or had "Super-Elite-Prestige" Hilton status or whatever.  

But you might even do better than the Apimondia rate if you have ID from a major multinational, or (hint hint) something that looks convincingly LIKE an employee ID from major multinational.  (I still hand over my AT&T Bell Labs ID card with my plastic when paying for hotels, and I am amazed how many hotels unquestioningly give me the "AT&T Rate", or their standard "corporate rate" even though AT&T sold Bell Labs to Nokia 2016, and I have not done an honest full day's work since 1993, when I "retired".)

Pro Tip #2:  Never select "Pay in [your currency]" when using plastic abroad, and presented with a credit card machine or bill.  Always elect to be (or insist upon) paying in the LOCAL currency.  The scam here is that you will pay a larcenous exchange rate that the hotel or their credit card servicing company imposes.  Most all credit cards offer a version of their card with no "foreign transaction fees", so let THEM do the currency exchange on YOUR behalf, as they will not give you worse than the "official rate".  Or pay in cash in Euros or USD dollars, as some places/people like hard currencies, and will discount prices sharply for "hard currency".  Do your homework, Chile is pretty stable - a high of $0.16 per peso, a low of $0.10 over the past 5 years, so likely no eyes will light up at the sight of your home currency.

EXCEPT for Argentina, where one wants to use a service like "Remitly.com", to wire oneself money in advance which has an exchange rate double the "official rate", and better than the "street rate" (the "Calle Florida rate").  Just wire money to yourself at Banco Nacional, and pick up a large wad of ARS pesos at the airport, as their airport branches are open 24 x 7 x 365.

And when dealing with government officials (like the "customs inspectors" that board recently-anchored boats in smaller island nations), always pay all fees by credit card or check, NEVER in cash.  I always say that we are on an expense account, and will not be reimbursed if I pay a fee in cash.  When sailing, we are never the ship's owners, we are just "a crew hired to reposition the ship for its owners", so we cannot pay for anything in cash, as we must account for every penny. So, you serve your best booze, you set out some treats, and have a nice chat with the "inspector", who might open a few drawers and cupboards, but never does the kind of job the US Coast Guard does, and he leaves to search for easier prey.

But the best way of all to save money is to learn more of the local language than your friends know, starting with "My friend here will be paying for the drinks."  😉

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