That does not look like you hit any plastic to me...
As backstory, I spent a long time as a high-end bicycle mechanic in NYC. Electronic shifting is and has been a thing for a while now. One system in particular (Shimano's Di2) has had a few big leaps in development, but in the earlier few years I had many clients inquire about ways to change or enhance and add functionality. As a result, I spent many, many hours digging potting compound out of delicate proprietary components in order to get to the center of the Tootsie Pop; in other words, to dig out a PCB from that same type of glossy black mess.
Most likely that entire steel (?) 'container' has been backfilled with potting compound. Vibration safety is often claimed, but it is usually just there to keep your hand out of the cookie jar (IE, telling the world it only is a $2 transformer available on Mouser). IMO/E, there is unlikely to be anything else in between you and any components entombed within. There might be a PCB in addition to the components, but I highly doubt you will encounter any 'shelf' like obstacle in there (other than the top of the transformer itself).
I would suggest using a heat gun on a very low setting rather than a soldering iron for this task. Or perhaps even a hair dryer. And don't linger. Get in and out quickly, and be prepared to do this over and over. Gather all sorts of Xacto knives, dental picks, balling tools and the like, and use the most appropriate tool as the landscape you uncover changes. Mount the whole thing in a table vise if you have one and dig very very slowly and patiently. And be prepared to rewire all the leads once you get to the finish line. By the looks of it, you've already hit a few of them with the iron, and may as well replace them. Since it doesn't work as is, all you have to lose is time... I would also suggest to start digging near one of the exiting wires. Once you find a lug, you will have a reference for the depth of the component itself and know to be careful going any deeper.
Alternatively, you could spend the same amount of time hunting down the part as subhuman has been very kind in providing you with the actual print (which hijackers hardly ever have available to them).
Good luck!