Remember that Kirk objected saying it would take too long, not that they would ever run out of fuel. And this is before they were aware of any Kelvan modifications.
We could chalk this up to there being many impossible things about the proposal, and Kirk shooting it down on basis of a randomly chosen one.
The same way we could say that a voyage to Sha-Ka-Ree is impossible for a dozen reasons, and Kirk quotes the Great Barrier rather than the distance because that nearby phenomenon is the first impossibility they will run into...
I'm sort of hesitant to accept the idea of Starfleet ships running on a fuel-less power system, even though we do lack references to things like refueling, tankers or fuel shortages (except when relating to impulse travel). Supposedly, antimatter fuel is involved, after all. And while antimatter isn't exactly a naturally occurring substance and indeed might need to be generated and re-generated, it would be a bit odd for the ship herself to be capable of doing that. If she can generate or re-generate antimatter at the rate the warp drive consumes it, why the need for antimatter in the first place? Why not hook up the generators directly to the warp engines?