Migrating database live while being severely handicapped
If you can, it would be better to do views and triggers rather than bilateral replication of data between both sides.

So I would modify your plan slightly:


Create a sane schema
Create an emulation of the old schema using views.
Create triggers on the views to write to the new schema instead of the old.


The question is what comes after this.  Since you are building a new tool, it is good to be in maintenance mode.  This means, I think, only fixing the slowest of the queries that cause you problems.