It’s possible, but unlikely that your issues are associated with having or not having a CUDA toolkit installed.
Professional applications (e.g. Adobe products) should recognize the GPUs they are capable of working with, and not require any toolkits to be installed.
This is entirely possible with CUDA. CUDA-aware applications can be statically linked against the needed CUDA libraries, in which case they only depend on having a proper driver installed for the GPU.
If a product like Adobe Premiere is not using your GPU when you think it should, I’m skeptical that installing a CUDA toolkit (or even a new driver, in the case of 347.52) would fix the issue.
The primary purpose of the CUDA toolkit is to enable developers to write code that uses CUDA. If you’re simply concerned about making Adobe Premiere work, I don’t really know much about it. My intent in commenting in this thread was to point out that the popup indicating the driver does not match your GPU is to be expected when trying to install an older driver or toolkit on a newer card.