After 3-4 weeks of the application, I received an email from the team leader that lead to a somewhat informal Skype interview a few days later. They were interested in my profile, which strongly matched the position, but offered an extremely low salary (32K-37K). They explained that this was very competitive for an NGO and that I'd be working “for a good cause”. I politely said that I was willing to go down from my current salary but needed some time to think about it, given that my current level in a publicly-funded job in the same sector was 55K.
Despite the salary disappointment, we left on very good terms, with a good feeling about the position match. A few days letter I sent a polite email to the team leader asking if it was possible to negotiate slightly upwards or, alternatively, that I would be interested in a part-time position with them (e.g. 80%).
They replied it was not possible to either negotiate the salary or the full-time position. This immediately led to being out of the process.
I was very surprised by the lack of flexibility showed by this company. In general, I would recommend not wasting time with an application, nor to fall into their false dichotomy - it is definitely compatible to work for a good cause and receive a decent pay!