That's where I was going with it earlier. I *think* it should work, because they are both serial streams, so once embedded should stay embedded. I don't know how channel mapping would turn out. But it is not low cost.What’s this gizmo? 4 channel XLR to SDI? Do SDI to HDMI converters then preserve MCH audio? https://www.bhphotovideo.com/c/product/1128870-REG/aja_3g_ama_3g_sdi_4_channel_analog_audio.html
However I also *think* (almost certain) it would require an SDI video source even if just a black sync generator. Additional cost there.
Black Magic makes a low cost $75 sdi/hdmi converter that works well with 2 channel, haven't tested MC. I wouldn't cheap out on the converter with other brands, some are sold for security cams.