Hi Neil,
just a suggestion what I'd try to do in your shoes in attempt to debloom manually with PixInsight:
1) Use RangeSelection to create mask covering saturated & nearly saturated pixels (image A)
2) Create suitable StarMask (image B)
3) Subtract B from A using PixelMath - the tricky part is to set up StarMask parameters so saturated star centers are removed from image A, leaving only bloom spikes
4) Select pixel math result as mask for your image
5) Try to apply MorphologicalTransformation, Erosion with horizontal kernel (assuming bloom spikes are vertical)
6) If needed, parameters in points 1-3 to create better mask
Of course this can't be compared with sophisticated inpainting techniques mentioned by Juan and you may as well use another application to do deblooming.
regards, Zbynek