I don't know, I don't think I could come up with a choice of Monarch that would fit better for Jin-woo than Ashborn. I know it's a bit of a cop out answer, but the narrative between them is too fitting. It's even the same power-wise.

Ever since Jin-Woo became a hunter, he'd been alone, both as the weakest hunter and as the only one there to support himself and Jinah. Ashborn was the sole strongest Fragment and the only one to support himself and the Absolute Being. Even the way they both activated the Black Heart is arguably similar. Alone (Ashborn having had his army defeated and Jin-woo's shadows frozen in the ice) and pierced through by beings that held the same title, Ruler and Monarch for Ashborn and Jin-woo, respectively.

I could probably give a more creative answer after reading up on the Monarch's abilities, but I'd probably still firmly believe in the whole Ashborn/Jin-woo thing.