Shared container (groupage) is 100% the move for a one-bed - paying for a full box when you've got sofa + boxes feels criminal. A mate did exactly this London → Vancouver last year, quoted three different companies and the prices were all over the place for seemingly the same service. Turned out the cheap ones didn't include destination delivery, just port.
She ended up going through removals to Canada from UK and said the transparency was the main thing - itemised quote, no port-only surprises. Transit was around 8–9 weeks which sounds long but honestly you're still setting up when it arrives anyway. Worth checking groupage slots book up fast for West Coast routes specifically. Get quotes early.