Mixing audio with HDMI video to replace the sound from the camera
It seems that what you are looking for is an HDMI audio embedder.

It seems that the Tascam DR-701D offers this functionality as described in Outputting this unit's audio as the HDMI output audio

Examples of external alternative such products can be found here (I am not affiliated with these manufacturers, other products might exist):

Kramer FC-69

Extron HAI 100 4K