I have nothing but good news for you,
My problem is that this is a remote position for an international company
This is the common and normal situation. All high-end development is "remote" and/or "internationalized", and always has been.
If you think you somehow make or pay less "because one party is in area X" - the good news is you are totally mistaken.
Thus, to repeat, you seem to have a drastic misconception. The good news: you do not consider at all irrelevant issues such as "what continent your current house happens to be on at the moment".
Hooray!
(A thought experiment - what would happen if you happened to move continents during the contract? Do you think you'd have to "tell them that and adjust your payments" - ?! Indeed, other than time zone, why would you even mention to them where you live?)
they are looking to pay lower than their local average which is too high for them
This sounds completely bizarre.
I'd walk away and forget about it.
(1) All programmers are very expensive (2) all software development burns incredible amounts of money (3) in the history of the universe, nobody has ever gotten any software project (from a little web site, to research projects) done on the cheap.
If they're trying to get "someone cheap" it just doesn't parse.
(In incredibly rare circumstances you can find a future-star new-programmer (who will very soon be as expensive as any other programmer) and get them cheap for a year. Other than that, why would anyone work cheap?)
On the other hand ...
My understand is there are "two different" figures you think should be the amount for the contract. (I am confused about which is which, as the question is not specific.)
The answer is:
Pick the higher of those two figures. Add 50%. Away you go.
To repeat:
In the actual question at hand which we are answering:
The country where you happen to live currently, is totally unrelated to anything. (Indeed, the same applies to where (if anywhere) the company happens to have offices.)
The they need something, "X", done. There's a market rate for "X". The market, "X", and anyone who can do "X", is completely uninterested in the location of the houses of the people involved..
Here's an actual recent example omitting names. About 8 weeks ago we had to hire for a company a top expert in a certain networking niche for a project. I dug up the best guy. (Sex anonymized here!) I told him $X. He said a higher rate, $Y. All I could really say was "OK". Money was sent around and work began. I actually didn't even know what country the guy was living in, until we were just talking socially once the project was underway.
How else can it be?