We plan to challenge the double-dipping in court and we are asking for the support of community to share this story in the hope that I will be picked up by the media.
I'll start off by acknowledging that copping two fines no doubt stings, and does seem (assuming the driver was respectful, courteous and acknowledged their errors) a harsh discretionary choice by the officer.
However, before challenging the double-dipping, note that a careful review of the video will show that there were two distinct offences here, which appears to have been overlooked by the OP.
The first we've all focussed on (Not give way) occurs at 0:12-017s, where the driver, with left signal still engaged (flashing signals are clearly visible at night with the M3 cameras), crosses lanes and merges right. Whether or not a collision was "avoided" is a matter of definition, but it is clear the police officer felt the need to take evasive action (headlights dip as front suspension compresses at 0:16s); immediately prior to this, based on the road markings and fish eye lens of the rear camera, the closest approach was roughly 4m, so the braking action appears well justified at the apparent speeds and with the unexpected lane incursion.
The second (Not give right change of direction signal with lights) actually occurs immediately after, 0:18-0:22s, where the right-hand turn at the subsequent intersection is made without indicating right (like the left hand signal earlier, the flashing right hand signal should be clearly evident in the right hand view, but is not present; the minimal flashing seen is from the police car's own right hand signal).... to appease the "he who is without sin" folk, I'll readily acknowledge that I was guilty of this on occassion in the M3 when making quick opposing turns before getting used to the behaviour of the turning stalk.
In terms of the claim of harrassment, on the evidence (and as evidenced by the general reaction here) I don't believe that will be a fruitful path. I fully expected to see a video where the police officer was tailgaiting the driver for several minutes before an inadvertant slip up (OP, if you have other video which demonstrates them following the driver for some time, by all means please post it). If this point is pushed, I suspect the magistrate will pointedly express incredulity that one can simultaneously be harassed by a police car and yet be so unaware of their presence as to cut them off.