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), and Ultra (4640), Insignia Roku TV, Sharp 4K Roku TV, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network. Samsung JU7100 4K TV, Yamaha RX-V583, CenturyLink DSL