According to Hyperdia, the distance from Nagoya to Kyoto is 247.4 km. So JR allows one stopover and max. two days.

The fare for a single "stopover" ticket (using Nozomi and Haruka) is 8,770 yen
The sum of the fares for two separate tickets (same trains) is 9,530 yen
The Nozomi fare + Kansai pass is 7,840 yen.

The savings from a "stopover" ticket on this itinerary aren't huge.

I think stopovers offer better savings for journeys on one express line that you break into two chunks (allowing you to avoid paying an express surcharge twice) than they do for journeys that involve two separate express trains. Thus a Nagoya to Kobe itin. with a one-night stopover in Kyoto will save you more money than a Nagoya to Kansai Airport itin. with a one-night stopover in Kyoto. (Corrected in the light of Joe's post below)

True budget travellers will take highway buses from Nagoya to Kyoto and possibly a local train from Kyoto to Osaka, followed by a Nankai airport train to KIX.