The Royal Navy converged on Bismarck beyond formal orders

Would Halifax have committed as many ships as Churchill did to hunt Bismarck after Hood sank?

Perhaps not, but much of the Royal Navy was already converging on Bismarck through local initiative rather than waiting for direct political orders.

Halifax might not have driven the pursuit as forcefully as Churchill, but the Royal Navy was already acting with its own momentum. A Tribal-class destroyer flotilla and several capital ships moved toward the reported position without being specifically assigned to the final battle. Their movement shows the depth of the fleet response after Hood's loss and the initiative exercised across dispersed commands.