Whoa..... slow the bus down.... Lets not start talking BECM replacements, resyncing and new keys, and all of this just yet over one window not working... Take a step back and have a look at how the system operates and it will help narrow it down!!! IF there is a problem in the BECM then it will be on the power board, and this can be swapped to keep the original logic board, which will keep keys, engine ECU all working happily.
The window switchpack talks to the BECM over a serial link - so all of the window switch signals are multiplexed down a couple of wires. The BECM then talks to the front door outstation over a serial link aswell to tell it what to do. If the mirror, and central locking work, and you get a 'click' from the relay in the outstation when you press the window up/down button for the offending window, then the switchpack, serial links, and logic side of the BECM are all OK and doing what they should.
You problem then is EITHER in the power board of the BECM (Supplying power to the fuse/output for the window motor), Wiring to the door, outstation, or wiring from the outstation to the window motor.
The front doors/windows are a bit different in that they have 2 positive power feeds going to them - one feed powers the outstation, central locking, and the mirror. The other feed is JUST for the window motor - and feeds into the outstation on the Purple/Blue wire. This feed comes from the BECM fuse F22 (30A) and is just for the window motors in BOTH front doors.
From the BECM to the door, the Purple/Blue wire goes through a connector in the door jamb (just inside the vehicle bodywork) which is known for getting water in and corroding.
If your other front window works properly, and F22 is intact, then iF the problem is in the BECM then it is likely to just be a burnt track on the power board (which would actually be repairable).
If your vehicle was in front of me, then I would do the following steps to troubleshoot it...
1) check for +12V on pin 12 of C1284 (99MY on) / C323 (Up to 1999) which is the 12 way GREY connector on the front of the BECM. This is the feed to the Purple/Blue wire that goes to the door outstation in RH door. If +12V here (should be permanently on) then BECM is OK, and move to next test. If no 12V here, then problem is on BECM power board, and it's time to pull it out and swap the power board over (NOT the whole BECM, otherwise that introduces a whole new world of headaches - keeping your logic board will keep your vehicle 'identity' correct - EKA, Mileage, VIN, Keys etc)
2) If 12v present at BECM connector, then check the connector in the door jamb. I've found a few where they've gone a bit green and furry. If there is +12V on the vehicle loom side, but not on the door loom side, then the connector/pins are at fault. If power at BECM, but not the vehicle loom side of connector, then there's a break in the wire between BECM and connector. If 12V present on both sides of the connector, then also check for a good connection (on a continuity check, not voltage for this one) between BOTH of the black ground wires and a the ground point in behind the kick panel, or to the BECM ground.
3) If 12V present on Purple/Blue wire in the door jamb connectors, and good ground (there's a 4th wire - Purple/Black in the connector in the door jamb - this should also show 12V between it and both ground wires - which we know it should as it's the power to the rest of the outstation - which we know works!) then check for 12V at the outstation on the Purple/Blue wire. If it's made it that far, then the problem is either the relay contacts/circuit board track in the outstation itself (and it's clicking but not doing anything) OR on the output side of the outstation in the wiring TO the window motor, OR the window motor itself... If you don't have +12V at the outstation on Purple/Blue, but you did at the door jamb connector, then the fault is in the wiring between the connector and the outstation.
4) if 12V is getting as far as the outstation, then put the meter probes into the connector to the window motor (Grey/Green and Grey/White wires) and on a digital volt meter, you should see +12V with the window switch in one position, and -12V in the other (the relay in the outstation switches the polarity to make the motor run in the appropriate direction). If you get +12V/-12V when the window switch is operated, then the fault is in the motor itself. If you don't, then the fault is in the wiring from the outstation to the motor, OR the outstation itself is faulty.
It sounds like a lot to test/check - but in reality with the door card off and access to the door jamb connections, it's probably about 1/2hr testing - and will help pinpoint exactly where the problem is occuring.
Hope this helps,
Marty