A story that has been picked up by a development team is too big. Is it 'okay' to break it down even though work has already commenced?
Yes, this is "Okay." ;-)
The Scrum Guide (scrumguides.org) actually encourages responding this way to emergence:
The Development Team modifies the Sprint Backlog throughout the Sprint, and the Sprint Backlog emerges during the Sprint. This emergence occurs as the Development Team works through the plan and learns more about the work needed to achieve the Sprint Goal. - scrum guide / sprint backlog
Additional points from the Scrum Guide
- The Sprint Goal is what guides the Dev TeamThe Sprint Backlog BELONGS to the Development Team
- No one tells the Development Team HOW to turn the product backlog into Increments of potentially releasable functionality
No, don't change stuff once work has started.
Even if you can split it without changing the requirements it will change the way it is implemented because you will be effectively adding interface requirements between the sub tasks.
Paint the fence red! Dev proceeds to buy paint sprayer
Paint this fence plank red. Dont change the colour of the pther planks! Dev throws away paint sprayer and scrubs paint off half the planks. Buys paint brush