If you used the Digital-Ocean 1-click install and not the Discourse official Standard Installation then you likely don’t have swap configured, so when you rebuild you run out of ram unless you have > 2GB.
You can try doing
cd /var/discourse
./discourse-setup
and it will create swap for you if it is needed.
If you want help with Digital Ocean’s one-click installation and want to “rely on DO folks to ensure reasonable behavior”, then you should rely on them to support you.
But since I’m already posting, an easier way to confirm that it’s not PuTTY that’s at fault (which might save a long time in fussing with PuTTY’s parameters for naught) is to try the console. If you haven’t run discourse-setup then I’m pretty sure that the issue is swap space.