The US Constitution provides that "When vacancies happen in the representation of any state in the Senate, the executive authority of such state shall issue writs of election to fill such vacancies: Provided, that the legislature of any state may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct."
So the default under the US Constitution is a special election to fill a Senate vacancy. In KY's case, the legislature, as provided by the US Constitution, gave the governor temporary appointment authority. But then the legislature, as within their power, repealed that law and reverted the law to the US Constitution default position.
Now, the argument may be made that the KY Constitution gives the Governor appointment power with respect to a US Senate vacancy. There are two weaknesses in that argument: One is that the US Constitution gives the authority to the legislature. The other is that the KY constitution doesn't expressly say anything about filling vacancies in federal offices. A court, in order to weave its way through the varying provisions, might very well find that the KY provisions only apply to state offices -- something that finds further support in that the KY constitution gives the governor to fill vacancies in "district" offices by appointment, but if that was applied to the US House of Representatives, it would clearly conflict with the US Constitution provision that requires special elections to fill such vacancies.
I'm not saying Beshear is certain to lose. Just that he is definitely not certain to win.