There is a procedure with BA to make an excess valuation declaration when checking your bags and to pay the additional fee; whether or not you can find anyone who knows how to do it will be another matter.
See section 15(e) of the CofC:
https://www.britishairways.com/en-us...ns-of-carriage

For example see:
https://www.britishairways.com/cms/g...ation-Form.pdf

The Montreal Convention actually provides for the making of the declaration by the passenger and the payment of the fee required by the airline in order to receive the benefit of the higher limit provided under the Convention for declared higher value baggage. See Article 22(2):
https:/www.iata.org/contentassets/fb1137ff561a4819a2d38f3db7308758/mc99-full-text.pdf