Closer inspection of the MultiDock's daughter boards reveals that the USB hub is connected to port 1 on the TS3USB221A. Per the datasheet, we should be able to tie the switch pin low in order to lock it to port 1. As the TS3USB221A is in a 10-Pin µQFN package, soldering directly to it would be a massive pain in the posterior. Luckily, the trace was reasonably easy to follow - a via right next to the pin drops it through to the back of the board, where it runs up to another via right next to pin 6 on the PIC. Each of the PIC's pins is almost as big as the entirety of the TS3USB221A, making it a much friendlier target for modding - and luckily, there's an ICSP header right next to the PIC, with a ground pin within easy reach.

Lifting the PIC's pin 6 from the board and soldering a jumper wire in its place, connected to the ICSP ground pin, does appear to have locked the TS3USB221A to port 1 - and one phone which didn't work with the unmodded MultiDock now happily works when plugged into the Orico hub via a modded daughter board. Unfortunately, that's not all that's needed; some devices seem to cause the PIC to switch something on the power rails (understandable, since it's trying to switch the port over to a dedicate charging port controller), resulting in the device's USB connection 'bouncing' continuously.

It's possible that the MultiDock's hub and/or daughter boards are broken, as I picked the MultiDock up second hand (well below full retail price) - and I've noticed a red LED blinking constantly on the MultiDock's main board, just beside the first USB2514B hub controller IC. With this in mind, and the lack of apparently quick-and-easy hack to make it work how I'd like it to, it's back to my original plan of removing the original electronics and replacing the whole lot.