Extending cache of generated GIFs from video

So, I have found out (through other posts) that yes, Cloudflare WILL cache thumbnails and GIFs generated from videos in Stream. However, it seems like these thumbs and GIFs (and poster images) are only cached for about 24 hours.
I would like to find a solution to extend this cache.
My use case: I show posters for videos on my webpage. On mouseover, it switches to a gif. Typical for many video related websites.
I have tried doing a cname for my domain to: https://customer-xxxxxxxxxxxxx.cloudflarestream.com, but cloudflare does not allow cnames to cloudflare.
I think if we could extend the cache times for these assets, that would be awesome. I would pay extra for it, and it would seem to help Cloudflare a bit as well, so new gifs don’t always have to be re-created over and over again.
Any ideas from the community?

Just for clarification, I have Cache Reserve and Tierd Cache turned on, AND page rules to cache everything on my website (cache level:everything, edge-catch:1 month)

Not being able to find a viable solution, I think the only thing left for me to do is download them and serve them from my regular (cached) folder.