The scenario you describe is not untypical and could be described as a "technical" offence. In the first instance I think you need to be asking yourself this:
Can you prove that you neither knew nor had reason to believe that there was not in force in relation to the vehicle such a policy of insurance at the time you were stopped?
Under the Road Traffic Act 1988 (sect 143) the onus is on you is to prove that you were insured at the time rather than for the prosecution to prove that you weren't.
If you are able to say that you have never received the letters sent out by Hastings Direct then I would suggest that you have a point to argue as a lawful defence but equally you need to be able to provide some form of evidence that you complied with the requirements to send in that proof of "no claims".
I have just had a look through the various pages on their website,the online copy of their car insurance policy and their terms and conditions and cannot find it written anywhere that you are obliged to provide written evidence to them of you previous "no claims". The only thing I can find relating to "no claims" is under the " Acceptance Criteria" but again that does not refer to any written proof or the need to submit it upon request. That said there are sometimes differences, for whatever reason I don't really know, between policies obtain via the phone and the internet and without physically seeing your paperwork it is hard to make judgement. You have obviously looked through what documentation you have so go to the Hastings website and see how things differ.
How did you pay for your insurance? Was it in one lump sum or in installments? If it was by monthly installments have any further payments been taken since that date of cancellation? If they have then there is again a potential to argue that you have been paying for a service and without having knowledge of your poicly being cancelled you have assumed that you were insured.
From what you write it sounds as if you were given a fixed penalty notice at the time. If you are going to argue the matter using the defence laid out in Sect 143(3) then you have the legal right to take the matter to court and do not have to pay the £200 or submit your licence for endorsement. If you surrendered your licence at the scene don't panic as the FPN acts as your licence in the mean time... so don't lose the ticket!
With regards to your car being seized the only way to get this back is to basically get yourself insured again ASAP and get it out of the pound. There is nothing that you can do other than pay this fee, it is not a fine but a statutory payment for the recovery and storage of your vehicle, but do keep all receipts and documentation regarding this as there is potential for claiming this back IF you challenge the no insurance matter in court and get found not guilty.
I think you need to look very carefully as to what has gone on and I would suggest you take full legal advice if you are going to challenge this matter in court. Even if you ask the matter to go to court but enter a guilty plea on the day you will atleast be in a position to submit any mitigation you may feel appropriate. Just be prepared that any fine you receive upon conviction is likely to be more than the £200 FPN as there will be costs incurred etc.
Let us know what you think... good luck!