"This is very interesting. What do you mean exactly by "I only need WU to feed data to my other site"? Would this be something we could offer you as well?"
My personal website uses Meteotemplate. It is a template created by Jachym. He is very active on this forum and extremely knowledgeable. My personal knowledge is very limited and I needed a lot of help getting my site/his template up and running. With that caveat, yes if you have a WU account you can feed the data to another template using an API from WU. The limitation IIRC is that you are limited to uploads every 5 minutes.
Another option may be to buy Cumulus by Sandysoft and modify it to add your site to upload to. Again IIRC the development of the program has ended due to the developer moving on to other things in his life. Someone, please correct me if I am incorrect on any of these points.