Is there a direct to PCB punchdown Ethernet connector?
It's not uncommon to have the magnetics external to the RJ-45 socket, especially in cheaper devices — Just like in this case, where I'm certain the magnetics are inside the TMSEMI package closeby. 

That board looks like it's only for 10/100 Mbps Ethernet; so I'm sure you could get by with nearly any PCB-mount punchdown connector there (preferrably rated cat 5 or better for 100MbE).

TL;DR:

Keep the magnetics package, and 'just' replace the RJ-45 socket with your punchdown connector of choice.



Edit:

Looking around, the magnetics part seems to be second sourced version of Pulse Electronics H2019NL/HX2019NL, a 100MbE, 15W PoE-specific part. Quite widely available too!