Sorry for the late reply but like @lassoan is saying your VTK/C++ should be able to be reused as-is. It is true that you will have to create a Python front-end to your code but with the wrapping available within VTK you should be able to drive your C++ code fairly easily and only focus on what the client (web/html) needs to trigger on the server side. You can even encapsulate all your C++ calls inside a single C++ class that will only expose in Python (via vtk/python wrapping) what you need from the web side.
A project that you may want to look at is PyWebVue (next gen VTK/ParaView-Web) and that VTK example doing remote rendering
The example readme suggest to use the code from the repo but it might be easier to use the published version.
virtualenv vtk-web
source ./vtk-web/bin/activate
pip install pywebvue vtk==9.0.20210717.dev0
python ./examples/01_cone/cone-vtk-remote-rendering/app.py --port 1234
HTH