The workaround that was suggested to us has been working well enough:
Search contacts for the customer’s email address.
Send an email via the customer profile (not through Conversations Inbox)
Go back to Conversations Inbox and mark the message Closed
The message will come from the individual user’s email address and their signature will be added.
Depending on how the message was received, it’s likely you’ll find the original message there and will be able to hit Reply so that the original text is included in the response. Watch that the correct sending email address is used because the default will be the shared email. But there is a simple drop-down to change that each time a message is sent.