Copying a project with templates

I have a project with 4 build templates at its top-level and then a large number of sub-projects and build configurations, each of which uses one of those templates.

In order to support a new branch I can copy this project easily enough, but all the copied build configs still refer to the original templates, implying that for each config I will need to disassociate it from the old template and associate it with the new one if I want to be able to modify the templates separately for each branch.

Given that there are ~90 configs in this project I'd rather not do this using the GUI, so can I use the REST API instead ? In particular, is it as straightforward as changing the template id for each build config in the new project from the corresponding one in the old project, or is it more complicated than that ?

I've looked at the approach suggested under "Howto Clone and rename Templates when copying projects"  but as I expect to have to go through this process a number of times each year I'd rather use a script if I can.