Does it take VGA? And do you have a spare PCIe slot in a desktop?
You can slap in an old GPU to power it directly, with no latency or conversion issues. I know my old Nvidia 980 TI supports VGA directly, albeit with a passive dongle. As a bonus, this would save VRAM for your other displays, and (if you wish) offload whatever you do on your CRTs from the main GPU.
Now, if you want a GPU with composite output, you have to go waaay back to the GTX 200 series or the Radeon 4000 series. Getting working drivers for these on linux might be an... adventure.