After trying Indy 4 Lazarus without success, I was told (thanks again, jesusrmx) to try Synapse's HTTP functions (from the HttpSend unit). I've been using Synapse for serial communication, but I was not aware of its HTTP features. I was gladly surprised: it's almost as easy as TIdHTTP and is officially supported on Lazarus.
Now I'll finally port the "Auto Updater" from my project. The preliminary tests were excellent, and as soon as I finish it, there will be another post here.