Insurers typically give you low-ball offers until the point you involve the Financial Ombudsman Service. The insurer knows you have no lawyer but the issue with instructing a lawyer is that for any claims under £10,000 in value, you cannot recover the lawyers charges as it would be a small claim. So most people pursue these disputes without any legal representation and indeed, there is a procedure to go through against the insurer.
First of all they are regulated by the Financial Conduct Authority (FCA) who can be reached on 0800(###) ###-####if you need to speak to them.
Next, as a result of the above, you must make a formal complaint to your insurer - the courts would expect you do this before you consider suing the the insurer.
You are covered under the Consumer Rights Act 2015 here which means you have a right to expect a service (from the insurer) carried out with reasonable care and skill. Furthermore, the policy should be “fit for purpose” and “as described”. If it isn’t then you can allege breach of your consumer rights and breach of contract.
You should now make a formal complaint and if that does not resolve the matter ask them for a "deadlock letter", which is a letter giving their final response. The insurer's complaints procedure will be available on their website.
Once you have the deadlock letter the next step would be to escalate this to the Ombudsman - they will investigate and liaise with the insurer which would hopefully result in a revised offer. They can order the insurer to make a financial award for inconvenience if they have acted poorly.
Once you have their final response, you can make the complaint here : www.financial-ombudsman.org.uk/consumers/how-to-complain
Or by email:***@******.***
Based on the circumstances, assuming the insurer does not uphold your complaint, I am sure the Ombudsman will do. The Ombudsman will look at this case independently and will make a decision based on what happened. Their decision is binding on the insurer so that could mean a higher figure to compensate you properly. If the Ombudsman did not find in your favour, therefore, you then have the option of suing the insurer for breach of your consumer rights and you have up to 6 years to bring a claim to the court (the limitation period is 6 years from your date of loss, to when you need to issue a claim).
The insurer is likely to be open to negotiation - you should show you are serious by going through the above though. As it often results in improved offers. I would not encourage you to instruct a lawyer as you won't get your money back and it would cost you a few hundred pounds at least if you did that so would be uneconomical given the figure in dispute.