The JER-HNL discount "through" fare will not cover that because the fare's rules disallow stopovers. The autopricer will have to find a different fare, or a different combination of fares to cover the sectors (provided the fares each permit the appropriate combination).

I never know whether the airline autopricer system thinks that JER is in the United Kingdom or not, but only "domestic" end-on-end combinations are permitted with the discount fare [due to an apparent error in the combination rules excluding intra-Europe fares offered by BA], so maybe you cannot even just add a cheap JER-LON to the LON-HNL discount fare.

Moreover, since your stay in the United Kingdom (proper) now exceeds twenty-four hours, the journey will also incur APD.