I have not tipped it over yet but been close so I feel your worry. I use only the rear brake to stabilise the bike when moving slowly - as much as possible with feet up. With feet down you cannot exert the same forces to balance the bike. I can manage almost full lock turns just by controlling...
I had a similar thing but not to the same extent. I was due to change the tyre so I did not pursue it with the tyre company. It did look like an issue with the tyre structure and not due to damage.