I would try to do it myself, were I you. Even if you don't speak the language you have plenty of time to do your research and get everything in order. There are some steps that must be done within a time period before arrival, and there is paperwork - but it's not insurmountable, you just have to go one step at a time.
What you were quoted sounds excessive. If you do it yourself there is very little cost.
Below is the whole process laid out (not including the airline accomodations you will have to make).
You can use Google translate, or Bing, or various others to translate Spanish to English and get reasonably accurate results.
First here is a link to find the Ecuadorian embassy and consulates in the United States. I would suggest you contact the Embassy or consulate nearest you and get any information you can from them - but you will still have to do what Agrocalidad requires. And I doubt they will help much, but it's worth a try.
This page from the government of Ecuador although oriented towards Europe, nicely summarizes what's necessary from Ecuador's perspective to import your pet(s) (not the airlines which is another thing you'll have to research by calling the airline of your choice):
From the above link, here's the summary of what's necessary to bring your pet(s) into Ecuador:
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¿CÓMO LLEVAR SUS MASCOTAS AL ECUADOR?
El propietario de una mascota (perro o gato) deberá presentar al inspector de AGROCALIDAD el certificado de vacunación en el aeropuerto.
Una vez que la mascota es inspeccionada, un certificado de salud con un valor de 26,17 USD se emitirá por el movimiento de los animales.
Para obtener más información, haga clic en el siguiente enlace: AGROCALIDAD
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Roughly, this translates to
HOW TO BRING YOUR PETS TO ECUADOR?
The owner of a pet (dog or cat) will have to present to the AGROCALIDAD inspector, the vaccination certificate in the airport.
Once the pet is inspected, a health certificate with a cost of $26.17 USD will be issued for the movement of the animals. [ ED: It is not made clear if this is for each animal, but probably is]
To obtain more information, click on the following link: AGROCALIDAD
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And here is that link to AGROCALIDAD:
Click on the green box with the text INGRESO AL ECUADOR DE PERROS Y GATOS.
That takes you here:
That page lists 3 things you have to do, paragraphs numbered 1, 2 and 3. There is also the second blue box with the text "Requisitos perros y gatos procedentes América" that you should click on after reading the paragraphs, which takes you to a PDF that details what vaccinations and certificates are necessary as well as the timeline requirements.
At the bottom of the page in the above link, there's a blue box with the text "*Procedimiento para realizar el pago del permiso zoosanitario de importación". Click on that box to see a PDF which tells you how to pay at various banks (but note it appears to be just a bureaucratic error that this document is titled for "exportación" not "importactión" and that the amount is given as $26.21 USD not the $26.17 previously specified). Note that this PDF also gives both the telephone numbers and email addresses to contact the Agrocalidad offices at various Ecuadorian airports.
So, that's it in a nutshell. This is involved, yes. But it's step-by-step and compared to what many of us deal with in our personal and business lives, not really a big deal. Is it $6000 USD worth of trouble? Not for me, YMMV. Do it yourself and you are out your time and legwork, the cost of vaccinations, and another $26 and change. I can't imagine any service to do all this that could take all the burden away from you - YOU will still have to get your cats vaccinated, arrange airline transportation, and make sure it's all done in the specified timeline.
Just another thing I should mention: The Ecuadorians are a fine people. But many of them will take advantage of the ignorance of foreigners, who are seen as a sort of walking ATM by some...so they will charge whatever the market will bear, and they have found that some gringos will bear almost anything...
Just an addition:Â Here's a link that lists airlines which fly to Quito.
LATAM has a page that talks about what's required to fly your pet(s) either in cabin or as cargo. I imagine other airlines also would have similar pages: