It sounds like there is an easy fix to this, which would be to exempt taxis, minicabs and Ubers from the ban when their destination is in the LTN. This could be done by allowing them to provide proof that they are a taxi driver and were taking a passenger to a destination in the LTN zone when the cameras pick them up and they are sent a letter fining them.
From reading the article, it sounds like your council doesn't want to respond to suggestions. But you never know.
Do you know your neighbours? Can you find other people who are concerned? It's better if there are a few people, but you don't need many. Most local councillors are worried about the effects a few voters can have, because most people don't vote at all or respond to 'consultations'. Those who shout loudest tend to get their way even if there aren't very many of them. Get to councillors' surgeries, or better still, neighbourhood meetings if they exist. (Covid might have stopped neighbourhood meetings though.) Can you find anyone who is disabled and affected by this?
You could practise your sad face and get the local paper involved. But be aware that pro-LTN keyboard warriors hate taxis, so if you do this you will get misogynistic abuse. (They all seem to be young, male, fit and selfish, with no concept that anyone might not be able to walk far or might need to take elderly relatives to appointments.) I hate to say this on FWR, but if you get some men onside it will mitigate the abuse.
Here's my example. A few years ago, our local council introduced a LTN that didn't include our street, but we were badly affected by displaced traffic. The council held regular 'neighbourhood meetings'. DH and three other people from our street turned up, for the first time ever, as did a bunch of people from another badly affected road, which was good because they could join forces. At the first meeting, the councillors were frankly rude - one played on her phone while people were making their case. But when they realised we weren't going away, the council actually made changes. Shouting loudly can work. Good luck.