Is it JUST fuse 1 that it blows? the Brown/Light green wire that is C105, on the post on the side of the BECM is one of the main power feeds to the BECM from the underbonnet fuse box. It feeds into the BECM and it feed fuse F1, F2, F3, F4 and some internal components on the BECM board.
So if you disconnect that wire at the fuse box end and it doesn't blow fuse 1, then that's likely because F1 isn't receiving power any more.
On my first P38, one of those main BECM feeds was pinched between the fuse box and the bodywork and rubbed through and was arcing to the bodywork and ended up frying the BECM for me... the previous owner had changed the fuse box, or so he told me when I bought the vehicle.
That Brown/Green wire, according to the ETM is fed from Max Fuse 1 in the underbonnet fuse box, and feeds the cluster, radio, clock, key illumination, and both front footwell lamps off of one feed wire from the BECM and then splices out in the loom. It also feeds the window switchpack and the rear footwell lamp from a different wire/connector on the BECM.
If you are just getting fuse F1 blow, then to me the issue HAS to be on the output side of the BECM, and most likely in the wiring loom. I would pull fuse F1, and measure which pin has 12V on it... which will be the feed from the Brown/Green wire. If you then se a multimeter and do a continuity test from the OTHER pin without 12V on it (as this is then the one that feeds out of the BECM to the splice and everything it powers) to ground. If you have a connection to ground that shows 0-1 ohms (depending on the multimeter) then you have a short to ground on the output side of the BECM.
If the vehicle has an aftermarket stereo in it, then that would be my prime candidate and would start by popping the head unit and checking the wiring in there. I've seen all sorts of issues stemming from bad connections, through to a self-tapping screw through the bodywork (to hold a LPG feed pipe that was running to the engine bay) which then hit into the loom and shorted a few different wires. I've also seen and repaired wiring issues where coolant soaked into the foam from leaking heater matrix O-rings has gotten into the wiring and caused it to corrode. One I fixed was easy to tell where the problem was, by the puff of smoke coming from the wiring loom when I lifted the footwell carpet.
I could be totally wrong, but if the fault was on the input side to the BECM, on that Brown/Green wire then a) you'd be seeing more issues than what you've described, as the Maxi fuse is rated at about 60A - and when that cable was shorting out on mine first one, it didn't blow F1 - it just caused bits in the BECM to shut down, and the cluster to go dead while driving, but the fuse was intact... and b) the fact it's JUST F1 blowing says to me that it's blowing as it's protecting something on the output of the BECM... If you know the BECM is good, and you get 12V to F2,3,4 with the Brown/Green wire connected and F1 removed, then your issue is further down the line.
I hope this helps...
Marty