Landships II

Sorry to answer my own question, but I've done a bit of digging and have found a few things that, put together, constitute something like an answer. I must admit to having resorted to The Great War Forum (sorry, Peter) where I received a response from an old friend of Landships who doesn't speak to us any more.

Anyway, the gist is that the long, pointed nightcap, often with a tassel, was widely worn by everyone in days gone by, and soldiers were no exception. In Napoleonic France they were issued to the troops, and many chose to wear them when not on duty or otherwise not required to wear the shako, bearskin, etc.

The style is referred to as à la dragonne. Dragonne meant, at that time, a sword-knot, so that presumably refers to the tassel.

The cap was initially worn with the tassel hanging down the back, then for convenience it was folded so that it hung on the right-hand side, and eventually it was tucked in completely, with the tassel hanging out at the front, above the eyeline. There are pics of Napoleonic grenadiers wearing the bonnet in its intermediate stage while in campaign dress. Finally it was  made in the simplified version, very similar to a forage cap, but still with the tassel.

Now the police bit. Men who were on a charge for something, confined to barracks or detained by the police, were required to put their headgear on when their punishment was announced, and the headgear they were most likely to have on them was the off-duty cap. Thus it acquired the nickname bonnet de police, even though it was worn in many other circumstances.

This slang term seems to have passed into everyday use. In bilingual Belgium it appears to be a purely French-speaking term, since the Flemish equivalent is the much more straightforward kwartiermuts, "barracks cap", just as some armies call it a "garrison cap".

Entirely by coincidence, the bonnet de police ended up looking very similar to other sidecaps, although its origins were completely different.

By the 20th century it had fallen out of favour in the French army, but was retained by Belgium until the 1930s and by Spain later than that. The Spanish Forign Legion still wears it today.

So that's it, as far as I can make out. I'm almost convinced.
 
Of course, the possibility remains that I've got this all wrong.

Pic of French bonnets enclosed.