If you need a consistent image for streaming, you should get one of those cheap SCART->HDMI adapters (and a SCART splitter if you don’t have a gscartsw with a second output).
It’s not the OSSC at fault per se; rather, it’s HDMI dropping sync because something in the video changed, either the resolution or the framerate. That causes HDMI to disconnect and everything in your chain to renegotiate. In order to avoid dropouts, you need a scaler with a full framebuffer that can both perform framerate conversion and continually output at a predefined resolution and framerate. Yeah, it would probably add at least a frame of lag, but that will be the tradeoff. (Use a CRT, play with lag, or play with dropouts–pick one.)
QMS in HDMI 2.1 should help to an extent by eliminating dropouts for framerate changes, so a scaler, like the Framemeister, that is configured to consistently output 1080p for all video modes, could be able to run without dropouts; however, I’m pretty sure that will need HDMI 2.1 on both ends, meaning new hardware revisions of the OSSC and/or Framemeister will be needed to take advantage of it.