Just to be clear, to log the time you must be 'rated', that is, you have a private or commercial certificate that says 'airplane single engine land'; and be the sole manipulator of the controls during that time. That's it. No TW endorsement is needed if it's a 9 instead of a 9A. Of course you cannot be THE PIC in that case, but you can log it as PIC time. One of the most confusing regs there is.
Insurance companies set their own requirements, which change all the time, depending on however they're feeling that day.
Finally, Falcon and Nationair are brokers, not insurance companies. It is their job to act as your representative. In principle they both have access to all the insurance companies, so in this case it sounds like Falcon didn't do their job very well. However, as to being paid, the fine print in virtually all aviation insurance contracts talks about 'earned premium schedule'. This means that the moment you say 'yes' you are obligated to pay a certain amount. If you cancel one day later you will get a partial refund but a lot less than 364/365. It is not pro rated.