Balloons can be quite useful for carrying scientific payloads, but only on Earth, Mars or something else "benign". Not Uranus though. In that context, it's more science fiction (of the remote future, that is) than a serious design consideration.
They are a poor design choice because:
- They are completely useless for the duration of the rest of the mission
- They involve tremendous risk if the mission's success is indeed to depend on them
- Using balloons would require very low injection velocities
- The must deploy at a certain altitude, because otherwise the pressure would be too low and the balloons would likely pop
That means you need a controlled and very slow decent into a volatile atmosphere, which costs tremendous amounts of propellant, reducing the possible payload mass. Any real space mission to Uranus would therefore likely choose something like controlled high-speed descent of the main probe, until total destruction (which will unfortunately not be very deep).
But, more to the point: Some work has been done no this subject, but not in much greater detail than a basic exponential atmosphere. Existing Uranian atmospheric models differ quite significantly from one another. As you may know, they are all based on Voyager 2 (1989) spectral data for the composition (aided by telescopic data). A lot of the current knowledge is inferred from analyzing the formation mechanisms and dynamics of the clouds in the Uranian atmosphere. This, together with pretty basic physics (like the exponential atmosphere and insolation models to get the temperature profiles) composes all of mankind's knowledge about the Uranian atmosphere.
Analyzing balloon flights on Uranus, given its inherent uncertainties, certainly does not require anything better than an exponential atmosphere. It also does not require you to use a full-fledged temperature profile for the integration of the barometric equation; just use the planet's average temperature (see the bottom of this wiki), and keep it constant; that significantly simplifies the integration. Then it's just a matter of plotting $P(r)$ vs. $g(r) = μ/r^2$ for a range of $r$ that describes the ~20% of the Uranian radius.