Not sure why your earlier comment got voted down. I voted it up to -1.

Could someone please explain any possible justification for exponential discounting in this situation?

I think exponential discounting has gotten ingrained in our thinking mostly for historical reasons. Quoting from the book Time and Decision: economic and psychological perspectives on intertemporal choice ("DU model" here being the 1937 model from Paul Samuelson that first suggested exponential discounting):

Samuelson did not endorse the DU model as a normative model of intertemporal choice, noting that "any connection between utility as discussed here and any welfare concept is disavowed" (1937, 161). He also made no claims on behalf of its descriptive validity, stressing, "It is completely arbitrary to assume that the individual behaves so as to maximize an integral of the form envisaged in [the DU model]" (1937, 159). Yet despite Samuelson's manifest reservations, the simplicity and elegance of this formulation was irresistible, and the DU model was rapidly adopted as the framework of choice for analyzing inter-temporal decisions.

The DU model received a scarcely needed further boost to its dominance as the standard model of intertemporal choice when Tjalling C. Koopmans (1960) showed that the model could be derived from a superficially plausible set of axioms. Koopmans, like Samuelson, did not argue that the DU model was psychologically or normatively plausible; his goal was only to show that under some well-specified (though arguably unrealistic) circumstances, individuals were logically compelled to possess positive time preference. Producers of a product, however, cannot dictate how the product will be used, and Koopmans's central technical message was largely lost while his axiomatization of the DU model helped to cement its popularity and bolster its perceived legitimacy.

I notice that axiomatizations in economics/theory of rationality seem to posses much more persuasive power than they should. (See also vNM's axiom of independence.) People seem to be really impressed that something is backed up by axioms and forget to check whether those axioms actually make sense for the situation.