It seems that what you are looking for is an **HDMI audio embedder**.

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

[Kramer FC-69][1]

[Extron HAI 100 4K][2]

You might check that the embedding process does not induce lipsync issues.

  [1]: https://www.kramerav.com/us/product/FC-69
  [2]: https://www.extron.com/product/hai1004k