I've been following this thread with some interest. Personally I'd RMA the board for replacement/repair, however, if that's not an option...

From the photo of the board it's feasible that some of the analogues to the left of the transistors are involved with the charge-pump - it's not clear but if I squint hard enough I can believe it. At this time I'd give-up looking for a "AA6-j" marked device - it isn't present in the top-hits of SMD-marking google hits. but the purpose of the device is clear - a logic-level N-channel MOSFET with a working voltage in excess of 24V with a commutating diode to avoid problems with back-EMF from the relay (this was one area that concerned me originally when considering this device could be a BJT). This would support the traces to OP1 being the only charge-pump output. That, in itself, is somewhat peculiar as I understand that the micro controller on the AXBB generates the charge-pump output to itself - so I'm not entirely sure of the logic behind this decision - the integrity of this safety feature is then at the lowest integrity of the board - likely the program running on the micro controller... but I digress.

In a bind, I'd look to sweat-off the MOSFET for OP1, and the three resistors for OP2, then haywire the gate signal from the landing pad from OP1 to OP2. This would transfer the charge-pump output to OP2. Or, you could carefully (without overheating) sweat off the MOSFETs for both OP1/2 and swap them - assuming that you don't require the OP2 output. It looks as though any/all of the OP1-8 have the same circuitry/options, with just OP1/2 shifted on the board to avoid the stand-off pillar.

Or replace the MOSFET with a generic with the same SOT-23 packaging and pinout - there's nothing special about it's function that requires it to be matched to the original. John's mention of the 2n7002 is a suitable device in SMD packaging.

John's analysis of the failed MOSFET is, I believe, the most likely scenario here. There is a question as to why this failed, but sometimes s**t happens. It would be worth, if possible, diagnosing this more completely before attempting a repair. OP has a meter, it would be sensible to measure the voltage at the "IC" side of the 220R resistor with the output OFF/ON (0V, 5V respectively) and at the transistor side of the 220R resistor (similarly, 0V, 5V respectively). That he mentions the LED turns on/off, however, it a pretty good indicator that this should be working.

OP asks about a sensible price for a repair shop to perform this - a blind replacement (they won't be able to easily test without a licensed UCCNC installation) would be 5 minutes work and 10 pence of component, but they run a business and have to cover their operating overheads. On a good day - price of a pint, however, if the boss is around I'd expect £30 upwards.

If the OP was closer I'd offer to help - but it's not really viable (100 miles). But if he wants to play with a hot-air rework setup with stereo microscope then the offer is there.