Hi there, my lo is also 12 weeks old.
About 3-4 weeks ago he would wake around 1am and then again around 4/5am. I treated both as night feeds, kept his room dark, and didn't talk or try to interact with him (although i was still gentle and cuddled him) and put him straight back down in his cot after he was fed and winded. After a while this meant that he would easily settle again after the 4/5am waking until around 7am. It was at this point that i introduced the dream feed at around 10.30pm. For the first few nights with the dream feed he still woke up twice before 7am, but was easy to settle, after about a week the dream feed must have started to work as he only started waking at around 5am. I still treated this as a night feed and so he was always easy to settle again. It is only recently that he has managed to sleep right through until 7am (although he has only done this twice), most of the time he still wakes at 5am or 6am, and i still treat this as a night feed. I never have any problems getting him to settle down again and i believe this is because i treat all his night wakings the same.
Your lo might resist it at first but it would be worth treating the 5.30am waking as a night feed even if he only takes a couple of oz, or a drink of water and then putting him straight back down in bed.
Hope that was useful to you.