There is a workaround that might work, but I don’t know because I’m not really a google docs user. This workaround might also be helpful for others - it didn’t exist when I wrote this request.
If you mouse over one of ChatGPT’s messages, some icons appear in the bottom left. The second of these copies the message to the clipboard, so if you paste it into a text editor you can see the whole message, including the LaTeX source.
If you have an appropriate plugin you might be able to get Google docs to render this. Note however that ChatGPT uses \( and \) for inline math and \[ and \] for display math. Depending on how LaTeX rendering works in google docs you might have to change these into $ and $$ or something else.