Firstly.please note that surface heterogeneity is not necessary for thermal convection. Convection happens because of the surface heat flux and thermodynamic instabilities, not because of heterogeneities. Heterogeneities may introduce secondary circulation and various currents.
According to the simplemost particle method, an air particle will raise roughly to the height where its temperature will equal the temperature of the surrounding atmosphere. I have no idea how much is 5.4 K per 1000 ft but the potential temperature is constant in the mixed part of the convective boundary layer and after the capping inversion with some step rise it will then increase with some gradient. If you heat your thermal say 1 K above the mixed layer potential temperature, then it will rise until it equalizes with the surroundings. If your pot. t. gradient above the mixed layer is 1 K / 100 m, the inversion is weak, it may rise roughly another 100 m. 1 K is already a lot. It won't get you kilometers above the capping inversion. Only large convective clouds can do that.
Even regular thermals do enter the stably stratified layer above the capping inversion and cause entrainment. This is the entrainment layer.
The real thermodynamic temperature is less convenient because it decreases in the mixed layer (9.8 K per 1 km) and then may either increse or decrase above the capping inversion (end still be stable stratified, the sign is not that important).
If your air is stably stratified with a constant gradient from the beginning, with no mixed layer, then you are probably describing the early morning transition moment you likely don't have that much radiation to oenetrate anything. During the day you should get some reasonable mixed layer. But let's say that for some reason you had some thick cloud cover with fog and what not and it suddenly disappeared.
Then you can approximately compute how high can a heated particle rise. If it is a really isolated heated source, it will work as a chimney with some plume rise.