Why is Uranus able to support a regular satellite system? Uranus has an obliquity of 98° which means that the mutual inclination between a satellite orbiting in its equatorial plane and the orbit of Uranus around the Sun would exceed the critical angle for Kozai oscillations. This would drive the satellites to high eccentricities which would likely be extremely damaging to the stability of the system, yet Uranus does host a system of regular satellites on near-circular orbits in its equatorial plane.

This suggests something is suppressing the Kozai oscillations for the regular satellites, so what is doing this? I suspect it is linked either to the non-spherical shape of the planet or gravitational interactions between the moons, but I'm not sure which would be a more relevant factor.
The solar perturbations on most of the satellites of Uranus are on a very small scale indeed, which may explain the absence of the instabilities noted in the question.  

A perturbational effect depends on the scale of the solar perturbing accelerations relative to the ordinary inverse-square attraction of the primary body. The scale factor (often designated $ m^2 $ in texts on the lunar theory such as Airy's 'Mathematical Tracts' and Godfray's 'Lunar Theory') increases with the separation of the satellite from the primary, it is also as the cube of the distance-ratio satellite-primary : primary-Sun (see detailed calculation below). 

Taking as an example the most distant of the major moons of Uranus, Oberon at 583500 km: 

The scale factor for the solar perturbing accelerations on Oberon in its orbit relative to Uranus is only about 1/5,000,000 of the gravitational acceleration towards Uranus felt by Oberon (calculation below). 

Comparing the corresponding scale factor for Earth's moon and its solar perturbations, the factor is very much larger, close to  1/178, as is well-known.  The Moon has a fairly strongly sun-perturbed orbit relative to the Earth, but the solar perturbations on Oberon are more than four orders of magnitude smaller, really very tiny. 

It is possible that the two tiny outermost satellites XVI and XVII of Uranus may be experiencing perturbations large enough to drive growth of their eccentricities, their eccentricities at 0.18 and 0.52  are very large compared with the major satellites, all with eccentricity &lt; 0.004 (source, Astronomical Almanac 2016).

Detail of the approximate calculation: 

The 'sampled' solar perturbing acceleration on a satellite is taken for present purposes to be represented by the perturbing acceleration on the satellite when it is close to quadrature with the Sun (as seen from the planet). In this configuration, the solar perturbation on the satellite is directed towards the planet, adding here to the ordinary inverse-square attraction of the satellite towards the planet. 

With mass-constants of the Sun, S, and of the planet, P ;  planetary semi-axis a ; and planet-satellite separation d , plus an assumption that the mass of the satellite is very small relative to the other two: 

1 ** the planet's accelerative attraction on the satellite is $P/d^2$ ; 

2 ** the Sun's accelerative attraction on the planet is $S/a^2$ ; 

3 ** and so also (to a quite close approximation) is the amplitude of the Sun's accelerative attraction on the satellite  $S/a^2$ . 

4 ** When the satellite is in quadrature, the resolved part of the Sun's vector attraction on it that does not cancel with the Sun's attraction on the planet, i.e. the net perturbation on the satellite at that point, is given very nearly by the proportions of the triangle satellite-planet-sun:  the lengths of two of the sides are approximately a and the third d. In this configuration, the resolved part of acceleration #3 in the direction of the planet, to a close approximation, is thus  -- 

$ (S/a^2) .  (d/a) $. 

The ratio of perturbing acceleration 4 to ordinary planetary attraction 1 is thus 

$ (S/P) . (d/a)^3 $. 

For the Sun and Uranus, S/P ~= 22902 , 

for the Sun and Earth+Moon, S/P ~= 328901 . 

Uranus is approximately 19 au from the Sun to the Earth's 1, the au is  149597871 km, the mean planetocentric distance of Oberon is 583500 km and of the Moon 385000 km.

Using these figures, the ratio 

'solar perturbation on satellite : planetary attraction on satellite'

comes to ~ 1/178 for the Earth and Moon, and  ~ 1/5,000,000 for Uranus and Oberon. 

The tiny outer satellites of Uranus are farther away from it than Oberon is, in a ratio of about 20.8 for the outer (XVII). Thus the perturbations on it, as a proportion of the ordinary attraction of Uranus at its distance, is $(20.8)^3$ times larger than for Oberon, making the relevant perturbation scale factor as large as about 1/550 , still smaller than for the one for our Moon, but perhaps enough for the disturbing effects to be reflected in its higher eccentricity of about 0.52.  

{Update:} It turns out that the orbits of the outer Uranian moons have actually been studied: Brozovic, M.; Jacobson, R. A. (2009), "The Orbits of the Outer Uranian Satellites", The Astronomical Journal, 137 (4): 3834-42. Just one of the (high-eccentricity) outer satellites was found to be disturbed by a Kozai resonance and may be in unstable orbit. It appears that the conditions for the effect are not fulfilled for the others.