The use of HTTP::CookieJar is recommended over HTTP::Cookies
Recommended by whom? Each of those modules has its own pros and cons and it is up to the user (coder) to choose between them given any particular scenario. They each have roughly 6,400 total dependents (according to MetaCPAN) so it isn't like one or other is dominating the scene.
If you want to use HTTP::Tiny for whatever reason then you have to put up with the consequences of that choice such as the manual translation between formats which you have shown. Hopefully it isn't too brittle.