Great questions KT. Hopefully I can help:
FIDI is the worlds largest professional association of removal companies, to be a member companies must hold FAIM accreditation. To achieve accreditation the company must pass a strict external audit by FIDI every 2 years which looks at over 200 defined quality standards including Financial stability, quality of resources (vehicles, packing materials, warehousing etc), knowledge and experience of staff, insurance claims, complaints and quality feedback, they audit the processes involved in international moving (For example booking processes, customer correspondence, freight management) and a whole host of other areas, basically their aim is to ensure that everything is in place to ensure a quality service is provided and that all loop holes are closed to reduce the risk of any problems occurring during a move.
The company I work for for example choose to be FIDI FAIM accredited to provide you with evidence that they we a quality moving company, after all, we can all build a nice website and send out a great sales person, but how do you know who to trust? By moving with a company that is FIDI FAIM accredited you aren't eliminating entirely the chance of anything going wrong, because moving is a physical process and there are some elements outside of the movers control, however, the if a company meets the strict FIDI FAIM standards (And ensuring any partners they use do too) you get peace of mind that these risks are as far reduced as possible. FIDI also provide expert training services which we use to ensure our staff knowledge and skills are kept up to date and are the best in the industry.
That being said, you don't HAVE to be FIDI FAIM accredited to perform international moving, it is the choice of the mover and there may be a good reason Crown are not. Why not have a chat with them about your concerns here and see what they say?
BAR Overseas is also important as they provide something called the Advance payment guarantee. This means that should anything happen to the company you book with during your move (i.e. they get into financial difficulty and shut down) that your payment is protected and that the BAR will arrange for another BAR overseas member to complete your services without you incurring any additional costs. This is only important for the mover you sign your contract with and as a UK association it's usually only the UK moving company that will be a member and not their overseas agent.
Hope that helps but if you want anything clarifying just let me know!
good luck with your move!