The last Roku that supported analog connections was the Express+. But that was last year's model, so finding one might prove difficult. Can't say why your HDMI/Analog converter won't work, but my guess is that it's also too old to support the 4K player. I just doesn't know how to handle 4K, and for whatever reason the Roku doesn't recognize that the adapter can't handle it and switch to a lower resolution.
I think it's more cost effective to replace your TV, rather than continuing to fiddle with converters. Depending on the TV size, it might not cost that much more than a quality HDMI converter, and then you'll have something that is capable of accepting anything with an HDMI connection. I bought a 43" 4K Roku TV for about $300 a year ago, and smaller sets are far less than that. Since you already have the Ultra, you don't need a Smart TV, reducing the cost more.
Nvidia Shield, Roku Stick (3600), Ultra (4640), Premiere (3920), Insignia 720p Roku TV, Sharp 4K Roku TV, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network.