As of June 2015 the paper counterpart for the UK photocard driving licence was abolished and no longer has any legal status, the government advice stating that the document should be destroyed.
Based on this, I wouldn't rely on the possession of the counterpart itself being taken as sufficient proof of entitlement for a number of reasons, for example, it no longer proves that your licence has not been revoked (because the government is no longer requiring it's surrender), or may not show that you have a recent serious driving conviction (because it is no longer being updated and re-issued following conviction).
Instead of the paper counterpart there is a new on-line system which you can use to provide your current licence details to relevant third parties such as hire companies.
https://www.gov.uk/government/news/driving-licence-changes