A fields parameter can take care of this.
In the example given in the question we could simply ask for the
customerIDfield.If the response provides a result then it exists.
From Best Practices for Designing a Pragmatic RESTful API by Vinay Sahni
Limiting which fields are returned by the API
The API consumer doesn't always need the full representation of a resource. The ability to select and choose returned fields goes a long way in letting the API consumer minimize network traffic and speed up their own usage of the API.
Use a fields query parameter that takes a comma separated list of fields to include.
Example:
GET /customer?fields=id,subject,customer_name,updated_at
Now, I suppose you could go the extra step and HEAD it, instead of GET
HEAD /customer?fields=id,subject,customer_name,updated_at
which should return a 204 if it exists & a 404 if not.