I believe you are asking two different questions: the first, how to use Coggan's Power Profile; and the second, given power at one duration, how to extrapolate and predict power at other durations.
But first, an historical aside: Coggan's Power Profiles pre-date the publication of his book with Hunter Allen so really should be thought of as Coggan's own. The Power Profile is an empirical summary: the top of each column is pegged to a "world-class" effort, the bottom of each column is pegged to a typical effort for an "untrained" but non-sedentary adult of normal health and fitness, and the range between the two were split into 40 equal-sized steps. The reason it was done this way is because Coggan lacked actual observed data on the true distribution of watts/kg in the population; he knew the top (="world-class" or "world-record") and he had a reasonable idea of the bottom but he didn't have a histogram of the actual percentages at each value of watts/kg in between. Importantly, the "category labels" were always considered to be just a labeling device to help readers understand the approximate levels -- Coggan had no specific data to support their semi-arbitrary assignment. In fact, in an early revision of the Power Profile Coggan removed the "category labels" because he felt that readers were mis-using them. There was great hue-and-cry and promises of good behavior and intelligent usage so he subsequently replaced the labels, though making them "overlap" rather than have sharp boundaries between categories as in his first edition. This is all preface to Coggan's intended and widely stated use of the Power Profiles: to identify relative strengths and weaknesses in a given rider's power. He intended that riders circle or highlight their watts/kg in each column, draw a connecting line between the highlighted values, then ignore the actual values and just look at the pattern described by the line. If the line sloped downward from left to right, it meant the rider was relatively stronger at short durations than at long, and either should work on "bringing up the right" or else exploit his (or, in later editions of the Profiles, her) skills as a sprinter. If the line was relatively flat, the rider was an "all-rounder." Importantly, the line was based on "best" efforts at those durations which would not typically occur during the course of a single ride but rather be pieced together from specific efforts from several rides. All of the modern cycling analytical software products can "extract" the best 5 second, 1 minute, 5 minute, and 1 hour efforts from your data. Some products can do so for any arbitrary length of time, for example 495 seconds.
So the bottom line for the first part of your question is that if you climbed a hill at 3.23 watts/kg for 8 minutes 15 seconds, you shouldn't use the Power Profile as a way to predict your power at other durations. The Power Profile wasn't meant for that purpose.
Which brings us to the second question: how, then, would one go about predicting power at different durations given power at one duration? This is a very common question but a full answer is complicated. That said, a simple answer (and the one that Coggan himself might advise) is to go out and ride a different duration and see what you can do.
The full answer from a modeling point of view is complicated because we have mainly two different "energy production pathways" to help us produce power (commonly though inexactly) called the "aerobic" and "anaerobic" systems, and we each fatigue and renew at different rates. Therefore, you would need several parameters in order to characterize a particular individual's power-duration curve, and there would be some (potentially, a lot of) error in the estimation. In short, at a minimum, you would need estimates of the parameters for "anaerobic" capacity, "aerobic" capacity, the actual work over time that caused fatigue, and for recovery from fatigue.
Coggan has recently hinted at what may be an empirical, non-modeling, approach. He has recently claimed that if given an estimate of a rider's FTP (functional threshold power) he can produce an estimate to within 5% of the rider's steady-state power at durations from 2 minutes on up.