In my mind, the most likely reason that you would not be able to form a diamond planet would be if the diamond formation happened in a region of the planet where it would not be able to coalesce into a solid surface. That is, if diamond conversion happened in a region where there was admixed non-methane ices, then the diamonds would be in a suspension or colloid in a volatile layer where, if your increasing-luminosity-sun hypothesis were to play our, the evaporating 'solvent' would carry away the diamonds.
Neptune's mantle is primarily made up of water, ammonia, and methane gasses. According to Bendetti, et al., 1999 (summary here), the experimentally determined conditions that will allow diamond formation from methane are 2000-3000 K and 10-50 GPa pressures. The question is: what will the rest of the water, ammonia, and methane gasses be doing at those temperatures and pressures?
Water and ammonia
Water and ammonia are both discussed in a Science report by Cavazzoni et al. The predicted conditions inside Neptune for the water-ammonia mix in the mantle implies that the water ammonia mix will be a molecular or ionic fluid, and will not solidify.
Also important is the density of these fluids. The same source suggests that water will have a density of 2250-2400 kg/m$^3$ and ammonia of 1400-1750 kg/m$^3$ at 60 GPa.
The other issue is with methane, or more specifically the byproducts of the methane breakdown. When diamond forms out of methane, there is obviously some hydrogen atoms that need to be accounted for.
According to the first paper, what first happens in the 10-50 GPa range is that methane 'fuses' into complex hydrocarbon polymers. For example, a reaction might be
CH$_4$ + CH$_4$ $\rightarrow$ C$_2$H$_6$ + H$_2$
Reactions like this proceed until enough of the complex hydrocarbon polymers start to lose their carbon completely, and form diamond crystals. This means that the released methane byproduct is going to be completely hydrogen.
The available evidence that I have found for methane suggests that at the given pressures and temperatures, it will be a supercritical fluid. For hydrogen, metallic hydrogen forms above the 500-1500 GPa range. While Jupiter and Saturn reach this pressure, Neptune does not. So both the methane and hydrogen should be supercritical fluids at the conditions that diamonds form in Neptune.
Unfortunately, I could not get any density estimates for either methane or hydrogen. However, both these fluids are generally less dense than either ammonia or water at any temperature and pressure conditions I could find.
The primary ingredients of Neptune's mantle should all be liquid. Furthermore, they should all have a lower density than that of a diamond at STP (~3500 kg/m$^3$).
Therefore, I consider it plausible that diamonds forming in Neptune's mantle would crystallize into a suspension. As the suspended particles got big enough over time, they would fall to the rocky core of the planet, coating it will a diamond surface. Up to 15% of Neptune's mass may be methane, so this diamond surface could be very large, as massive as the Earth, and possibly larger in diameter due to its lower density.
If Neptune's atmosphere and upper layers were stripped away by a more luminous sun, then it is plausible that a diamond surface would be exposed. However, I will say that with the outer layers, some 80-90% of the planet's mass, removed, the solid core around which the diamonds formed would undoubtedly expand. This expansion would probably shatter the diamond, which would no longer be under enough pressure to reform. So, if the diamond planet were true, it would be a shattered diamond wasteland, instead of a smooth crystal planet.