Advice to partial rename uri path

That's some very specific stuff you're trying to achieve .. 

If the "client URL" you want to display does not exist on the server, I think you'll need two instructions in the iRule, one to redirect client to target URI and one to change this string before it connects to the webserver. 

You might try something similar to this

when HTTP_REQUEST {
  if { [HTTP::host] equals "companyA.com" }{ 
    #I'm using a switch statement so it's easier to add/change uri behavior in the future
    switch -glob [HTTP::path] {
      "/" { HTTP::redirect "https://[HTTP::host]/bed/table/r/Home" }
      "/bed/table/r/Home" { HTTP::path "/room/desktop/r/Home" } 
    }
  }
}