Why does nitrous oxide have 300 times the global warming potential of CO2?
There are two really important properties of N2O that make it such an important greenhouse gas, both of which you have raised in your question but need some clarification.  


Infrared absorption - it's good to think in terms of gaps here. There's a lot of overlap between CO2, CH4 and (a frequently ignored GHG, but very important) water vapour in the wavelengths which they absorb. At the 4-5 micron and 7-8 micron range, N2O is extremely efficient at absorbing infrared radiation. These are important regions, precisely because there is no overlap with other GHGs in these regions. N2O is absorbing infrared very efficiently, and is doing so without "competition". 
Lifetime - you noted that the lifetime of N2O and CO2 are comparable, which is sort of true. N2O, as a stable, inert molecule that is well mixed in the atmosphere, has a very long lifetime of 120 years compared to some other GHGs (CH4, for example is 8 years, based on the fact that its removed by the hydroxyl OH radical). It's possible to determine this fairly accurately as the only sinks for N2O are its stratospheric photo-dissociation and reaction with O(1D) radical. CO2 by comparison has a lifetime that varies anywhere between 5 and 200 years. The comparable part comes from the fact that for calculations for global warming potential, CO2 is given an "averaged" lifetime. 


With N2O, therefore, we have a very efficient infrared absorber at two "windows" at which there is no overlap with other GHGs, and a long lifetime. The Global Warming Potential is the time integrated radiative forcing for a 1 kg pulse emission of the compound, where the upper limit of the integration is time horizon (usually set at 100 yr), relative to the same quantity of the reference compound CO2. It becomes clear from this definition why a stable, inert molecule with a long lifetime and which is very efficient at absorbing at particular infrared wavelengths would have a greater global warming potential than CO2 (shorter lifetime, absorbing in "windows" shared by other compounds) on a kg to kg comparison.