I think there are 2 alternatives that I can think of:

Alternative A

First, when there’s a new incoming email, do a Get contact with email_fromAddress variable as a key. Then you’ll get an Id of contact record

Second, do a Get Case with id from first step, if there is possibility a contact has more than 1 case, you can select the case that has flag e.g survey sent

Third, do Update case with id from second second step

Alternative B

First, you update your sendbetteremail subject template to e.g: “Survey from Chad ref_abcd” ref _abcd here means your external id of the case.

Second, setup the flow to clean/parse subject of the email (so you’ll only have an externalId) and then query the case you wanted with that externalId

Third, update your case from second step

Anyway if you’re still have difficulty about this, and want to discuss more I’d be happy to help via email