You can't just mess with the metadata; you have to actually remap the image.

Flickr's panorama interactive viewer can only be used with equirectangular 360x180 panoramas. And to make it interactive on Flickr only requires adding the "equirectangular" tag (unlike Facebook, where you have to set the metadata). But, the image does have to be an actual equirectangular projection that covers the entire spherical view. So, filling in the missing floor/ceiling areas with black to get a 2:1 rectangle is possibly your best bet. You could try using Hugin and loading up your panorama as a Panorama (cylindrical) "lens type" and then remapping to an equirectangular 2:1, with steps similar to those in this Q&A.

Flickr's interactive viewer does not recognize 360º cylindrical panos and does not make them interactive. The only way I know of to interactively view a cylindrical 360º stored on Flickr, is to use a viewer that that uses Flickr's API, such as Aldo's fieldofview.com. To use it, create a URL in the following format:

http://fieldofview.com/flickr/?page=photos/user_ID/photo_ID/&tags=TAGS

Where user_ID is your Flickr user ID, photo_ID is the photo's ID, and TAGS are any other tagged photos you want to throw into the viewer.

So, for example:

https://fieldofview.com/flickr/?page=photos/l0b0/44945085211/&tags=360

gives you an interactive view of the pano, with a linked gallery of everything tagged '360' in the user's Flickr stream. But it assumes that all panos have 360º coverage, so the left and right edges are going to be wrapped around and joined together in the viewer.

See also: Wrapping photos to an equirectangular projection in Hugin.