We recently implemented this for one of our REST platforms and we used a combination of BOTH the points you mentioned, meaning Authorization header & JWT token. Although, JWT is ONLY for authentication and GETTING an access_token (oauth token) which is later used with calling actual resource apis. I will discuss how we handled this situation and you can decide on how you want to implement it.
1) Authentication
Client sends a JWT to your authentication service (/api/oauth2/auth). (If you want more reading on JWT, you can read here and here of how JWT is implemented by google and how you can use spring-security-jwt libary to handle all the signing and encrypting/decrypting). You get the "clientId" out of JWT after decrypting and verifying the signature and after server does all the authentication, you respond back with a 'refresh_token' and an 'access_token'. Server will save the access_token as well and map it to the clientId so that when client makes requests using access_token, you can know which client is making the request. The access_token expires in some time (ideally in an hour) and when it expires, the client uses the 'refresh_token' to get a new access token by posting refresh_token to some refresh token url (/api/oauth2/auth/token)
2) Authorization
Client takes the 'access_token' and uses the access token to make all the subsequent requests on all other apis (/api/*). Ideally, the access_token is sent as a part of the "Authorization" header. Server uses request filters (if you are using JAX-RS, you can use something like ContainerFilterRequest to add filters to specific url patterns and intercept them) to filter EACH request and parse out the Authorization header value. You will get the access_token from the header and from the access_token you can get the clientId that you mapped in step 1). You can do other authorization logic in the security filter and if everything goes through, you can use this information to LOG that clientId and the request that the client made.
This way you can kill 2 birds with one stone : Implement a security layer & log the information about customers (what calls they are making, how many time etc. etc.). In case you don't want to implement security filter just yet (as you mentioned it might be in the future), for now, the clients can just pass on the "clientId" (base64encoded or not, upto you) as a part of "Authorization" header. If all the calls are from a "trusted" network, it should be ok, although not as secure. This way, when you ACTUALLY implement a JWT and Oauth based security layer, all you have to do is change your ContainerFilterRequest logic to parse out access_token instead of client id (as mentioned in step # 2).
I hope this helps ! For more information on security filters you can have a look at this answer: Basic Authentication of a resource in Dropwizard. It says dropwizard, but it mostly talks about JAX-RS.