I cannot see that there is anything to claim from BA

The 1st ticket was purchased via a TA and then changed by the TA. The TA failed to complete ticketing ( which is its responsibility )

Due to this error, you were unable to travel since you had been a no-show for flight as far as BA was concerned

You then purchased another ticket for which there is no reason to expect a refund

The company to be claiming the cost of the new ticket from is the travel agency that messed up

I do not see what grounds there would likely be for an insurance claim. Insurance policies do vary, but would be surprised if a travel agency not issuing a ticket would be one of them