Does displaying HMAC hashes of user inputs compromise the HMAC validation?
Does allowing users to test VALUEs increase the likelihood that SECRET will be broken or illegal hashed values generated, relative to the CONTROL scenario? 


By definition a cryptographic Message Authentication Code such as HMAC is secure only if resists existential forgery under chosen-plaintext attacks. i.e. if allowing users to test VALUEs increases the likelihood of discovering the SECRET or generating illegal hashes, then HMAC would not be a secure MAC (and the current state of our knowledge is that it is strong when used with a half decent hash function).

In your particular usage though, unless the HASH value is tied to a particular cart/session it might be possible to replay HASH values obtained from previous interactions.


  By how much?
  How is this calculated or proved?


The 2006 security proof for HMAC by Mihir Bellare provides a proof and tightness bounds on that proof, but for your purposes you could probably treat the difference as negligible.