You obviously don't need passports to travel within the UK as far as Border Force officials are concerned. However Loganair does require photographic ID when travelling on domestic flights with luggage in the hold.
Quote:
"Government-issued photographic ID is required on all Domestic flights when checking in luggage. Here's a few examples of Government Issued photographic ID that is accepted when checking in Hold luggage:
* A valid passport - an expired passport can be used up to a maximum of five years after expiry
* Valid photographic driving licence, provisional or full
* Young Scot Card
* Bus pass
Please ensure you still look like the photograph in the ID you are presenting."
Source:
https://www.loganair.co.uk/travel-help/id-requirements/
I've looked into the possibility of travelling from Basingstoke to Inverness by rail, and then hiring a car. The prices aren't too bad (totalling around £300 return for the three of you for the rail journey) but the timing don't work out for the ferries from Ullapool. They also don't work out for getting a flight from Inverness, rather than hiring a car. So that seems to be no use. Going by rail to Glasgow or Edinburgh, and then picking up a flight to Stornoway from there, doesn't work out due to the timings either.