Construction of the waterfront museum was beset by problems in relation to a flagship steps and seating area, as well as ceilings.
The museum board took AEW Architects & Designers Ltd to the High Court earlier this year in a bid for damages to compensate for the problems and pay for remedial works.
Last month, it was awarded £1.1m to compensate for some of the problems and stands to get another £1,236,419 after today's decision.
Mr Justice Akenhead said today's additional payment was due for faults in ceilings, one of which partially collapsed after it was installed.
The judge said some of the damages - just over £200,000 - were to cover staff costs relating to the logistics of storing and caring for about 3,500 high-value artefacts, including around 800 which are on loan to the Museum, while repairs are ongoing.
He said he had heard in evidence these artefacts include a pair of John Lennon's spectacles - regarded as 'particularly iconic' by Beatles fans and worth around £1m.
These are on loan from the V&A Museum in London and will have to be returned there by hand, by a trusted member of staff, and collected in the same way once the works are finished.
The central problem with the museum was an area of the amphitheatre referred to as "the valley" which contractors thought was unbuildable.
Instead, they installed a "plinth", which museum bosses described as an "abomination", leading to its removal at significant cost.
AEW admitted fault for the ceiling which partially collapsed and the judge ruled it should have realised another type of ceiling was "potentially dangerous".
The Manchester-based architects have already secured a £205,080 contribution towards the total bill from contractors building partnership PIHL Galliford Try JV.