3. Cat 3. itself didn't require the nose up trim input, which is a dual autopilot issue, where it has to achieve a fail passive outcome, If a 3rd AP had been added, 3rd power source, independent instrument transfers etc... then fail-operational would have been possible which does not involve a nose-up trim bias input. B747s, triple or dual autoland could be conducted, Triple avoided the out of trim if a manual landing was made with a late AP disconnect. The B737 dual AP has to be fail passive, hence the trim bias.
The 737 is perfectly capable of fail operational CAT IIIb operation. It is just that most carriers do not want to spend the extra money on the maintenance for the rudder servo that is fitted additionally. The autoflight system is inherently capable of CAT IIIb out of the box though, it is just not activated for most carriers. And the CAT IIIb fail operational airplanes still have the same trim up thing going on, both in dual engine and one engine out autolands (yes, those are approved as well if one has the fail operational version, flaps 30 for landing as well).