Back to answer my own question in case it's helpful to anyone else. I found a company/broker called Clegg Gifford to insure just the car, so we have ended up with shipping by Pickfords, Contents insurance by Letton Percival and car insurance by Clegg Gifford. Each party is aware of the others' involvement.
Having dealt with all the paperwork, it's now time to start the cleaning. We seem to be doing a cupboard a day and in every one there is something to trip us up - a grass encrusted picnic rug, a jar of beeswax polish. Probably worrying too much but better safe than sorry!