Proxy subdomain content to subdirectory on root domain? Is this even possible?

Hey!

Sorry, bit of a technical question here, but I'm sure others have encountered this as-well.

I have a site. Right now the domain structure looks like this:

mysite.com => landing page w/ unicornplatform
blog.mysite.com => ghost blog
widgets.mysite.com => micro-tools built w/ React & embedded as iframes in root
docs.mysite.com => help center on engine.so

I want to re-arrange the structure so it's like this:

mysite.com/tools/<any> => proxy content from widgets.mysite.com
mysite.com/blog/<any> => proxy content from blog.mysite.com
mysite.com/docs/<any> => proxy content from docs.mysite.com
else: proxy content from my current landing page

And for some reason this is very very challenging. I'd assume I could just setup a reverse proxy, but there's SSL and relative links and whatnot that always break. Why is this so hard and is there any solutions (even paid ones)?