I finally managed to make it work, using PuTTY’s Pageant authentication agent instead of ssh-agent, and following the steps mentioned here (it’s for Visual Studio Code, but works for Visual Studio 2017 and I guess it should work for any application that uses the “official” Git for Windows).
Since I already had Pageant installed and a .ppk private key created, I only had to make Git use Pageant, by creating the GIT_SSH Windows environment variable and setting it to the path of the “plink.exe” file (inside the PuTTY installation, for example C:\Program Files (x86)\PuTTY\plink.exe). With that done, I just need to open Pageant and add the private key (and leave it open while working with the repository), and Visual Studio will be able to connect and issue commands just fine.