Today we got in touch with a boat broker I visited last year, who are specialized in liveaboards in Marina del Rey. I gave him a list of boats we researched on the main website where boats for sale are listed, YachtWorld. The list contains some thirteen boats that fit our criteria: between 43 and 53 feet long, diesel engines, motor yachts as opposed to sail boats, and some other things like multiple living spaces and at least two cabins. Here's an example of such a boat:

We also booked a flight for November, to look at all of these boats, and hopefully narrow down the search to one or two that we can have a survey done on. The actual purchase could be done once we are back in France, 'long distance'. The aim is to have the boat ready and waiting when we move to LA, in January (if all continues to go according to plan).