This question needs to be taken in two parts, one relating to authenticity and the other to Kant's rigorism about lying.
Kant does not use any term corresponding in sense to 'authenticity'. But this does not mean that he has nothing like a concept of authenticity. One can have and use a concept without having a specific term for it.
A concept of authenticity is present in Kant. Briefly, Kant recognises that one can be true to one's rational nature. This means (a) treating oneself and not only others as ends; it also means (b) acting on self-legislated, universalisable moral rules, and (c) overriding 'heteronomous' motivation - motivation induced by external constraints imposed by others or acting on mere impulses of nature. It is very natural, whatever Kant's language, to say that
someone who observes (a) - (c) in their life is being faithful to their rational nature and in that sense, according to Kant's view of persons, is acting authentically as a rational being. Whether we accept Kant's view of persons is another matter.
Kant's rigorism about and against lying cannot be denied. It is obtrusively present in the Groundwork, in the Critique of Practical Reason, in the Metaphysics of Morals and in the essay 'On a Supposed Right to Lie'. It makes Kant look like a moral dogmatist, inflexible and insensitive to the complexities and nuances of the moral life.
A methodological point should be noted, though. Whatever the strengths and merits (or otherwise) of Kant's ethical theory, Kant is no infallible or even privileged authority on its applications. His ethical theory does not necessarily imply what Kant thought it did, especially concerning 'the madman at the door'.
A universalisability criterion works (sort of) against borrowing on the basis of a lying promise to repay. If everyone borrowed on this basis, then no-one would lend : lying would be counter-productive and destroy the very institution, that of lending, which it exploits. (I say 'sort of' because, while it would be highly odd if lending continued under these conditions, it is not logically impossible for it to do so. It is not logically impossible for the maxim, 'lie whenever you want money and can get it by falsely promising to repay', to be universalised.)
Now, about lying to the murderer at the door - there are two moves we can make against Kant here, both derived directly from his ethical theory.
The first is that in the Metaphysics of Morals Kant tells us concerning duties to ourself : 'The negative duties forbid man (sic) to act contrary to the end of his nature and so have to do merely with his self-preservation' (6 : 418 : M. Gregor tr., 1964). So there is a duty of self-preservation. This duty may require me to lie as the only effectual means of fulfilling it. If I may lie in order to secure self-preservation - perhaps my life is at risk from unjust aggression* - there is no relevant difference between myself and another (the murderer's potential victim) which forbids me to lie in order to secure their preservation. The maxim, 'If I can preserve another person's life from unjust aggresssion only by lying, then lie' is perfectly capable of universalisation without contradiction. It involves no logical impossibility of universalisation.
The second is that if Kant still insists that we have a duty not to lie, then we have a conflict of duties: in order to be fulfilled the duty of self-preservation (logically extended to another person) may require a lie, yet moral rigorism imposes a duty not to lie. But Kant denies the possibility of a conflict of duties; it is indeed implicit in his 'ought implies can' principle. If two or more duties conflict, they cannot both be fulfilled but as duties we ought to do them. The situation is incoherent.
Agreed, it remains open, if we have a duty to lie to the murderer at the door (as I have argued) and a duty never to lie (as Kant repeatedly insists), which duty has precedence. But Kant gives no grounds - rationally or morally - for giving precedence to the 'never lie' rule.
- The phrase is Michael Cholbi's in 'The Murderer at the Door: What Kant Should Have Said', Philosophy and Phenomenological Research, Vol. LXXIX No. 1, July 2009, 17.