Magento – Duplicate Sub-sub categories

If it's too complex, than fake the feature.

1) You could try to have a multi-select attribute. The options in admin should be padded. Padding spaces are levels to that node. Like categories in multi-select but with +1 space before childs categories. Or you can have a custom renderer in admin for that attribute which is not hard to make. On frontend you can render in layer navigation all values in a html/js tree. Values can be expandable and selectable. The values of this multiselect are searchable. If you want urls for this attribute have an extension that creates attribute_value.html pages like 'Shop by Brand' extensions. It will be a hassle to customize it for your exact needs. A variation of the attribute solution is to use a custom source table. In this table you can have a hierarchy of rows defined as you see fit.

2) A real solution would be to define a whole new module with a new EAV entity. Try to use Varien_Data_Tree. Plenty of examples in Category module. You'd not have to duplicate Category module entirely. This implementation is very complex in my opinion. It is a Magento developer nightmare - mine at least.

3) A simpler variation of 3) is to have a new entity (not EAV). Have it like the classic way of creating categories in sql tables. (parent_id, child_id) table or a column with parents of the current row. Products will be linked through an additional table with the taxonomy entity (taxonomy_id, product_id). Do use indexers and do cache most of the blocks for this taxonomy entity. Their data won't change too often.

4) I don't see customers using two trees that are in fact 2 different ways of reaching a product. The ones that don't know what to look for will have a harder time navigating in both trees. I guess you can have 2 'main' categories. One will be used in navigation and the other for 'taxonomy'. For all categories products ca be assigned. You'd have to make sure to use in navigation and layered navigation only the categories from the first 'main category'.

People that take decisions lose touch with reality now and than. Magento is a complex eCommerce platform. Merchants tend to bring their complex business logic in customers minds. Magento is a complex eCommerce platform. Implementing an extremely complex feature takes time and resources. A successful Magento store is a fast one and easy to use. If you approach this matter they'll come to their senses. Look for a different feature that addresses the main concern and that would replace the original feature (taxonomy tree).

The category move is an expensive event.
It triggers the reindexing of url rewrites and category product association (and maybe others).
You should put the indexes on 'manual' instead of "on save", move your categories around and reindex everything when you are done.
I'm not sure that truncating core_url_rewrite table will solve your problem.
It may solve it for your first category move, but you will end up with a lot of rewrites again.