This sounds like a really difficult situation. It can be so frustrating when people don’t pay and can impact on us both in terms of the time we spend chasing and the emotional journey it sends us on. I think in general our clients are pretty good at paying so it comes as a surprise when someone refuses. It sounds like there is an issue here with a difference in yours and her perception of the issues for her daughter. Sometimes it’s possible to make some changes which supports a parent’s view without compromising our own professional judgement but there is a point at which we may have to agree to disagree. If she has paid for your professional view and doesn’t agree with it and you feel that the standard of your work is as you agreed then the fee should be paid. Even though she has not signed the Ts and Cs you still have a verbal contract which should stand up. As I see it you have two options here and you will need to consider the impact of each on your time and personal well being.
Cut your losses and see it as a learning experience, make the changes you’ve already identified to your processes and hopefully it won’t happen again. You will lose some finance and be left with some residual feelings of anger/frustration but you won’t have to spend any more of your expensive time trying to chase this or deal with the situation. It won’t feel resolved but if you can live with that discomfort you can crack on with putting time into future work that will continue to bring in money.
Continue to follow a process to claim your fee. You can use the small claims court and this should cost you around £35. However, it will cost you a lot more than that in terms of the time that you put in but may be worth it if you end up with a feeling of resolution (as you say the principle is important here and this may be more important to you than the time factor).
Either option will have a cost for you and only by thinking about what matters most to you will you know which route is the right one to take.
There is a possible third option which is a compromise between the two.
Tell the client that you intend to make a claim through the small claims court. Ensure that they are aware of the implications of this for them i.e. if you are successful and they receive a county court judgement against them. In this case they are unlikely to be able to get credit for 6 years (this may include mortgages, car loans, paying for anything in installments). Indicating that this is what you are prepared to do if you don’t receive payment may be sufficient to prompt them to pay. It is then up to you whether you continue down this path if they still refuse. Here is some information related to this: https://www.citizensadvice.org.uk/debt-and-money/borrowing-money/county-court-judgments-and-your-credit-rating/
Perhaps this is one to take to a supervision session as there are a range of things to consider and it can be helpful just to offload about these kinds of situations. In the meantime I hope that the above gives you some things to consider.