The Rogers article is impressively analytical - but it assumes a constant rate of bank - a simplification which costs you in this special situation.
I experimented with this at some length in a single many years back and determined that (under the prevailing circumstances) one could gain considerably in performance by using an intentionally non-uniform bank/speed profile. To do so requires a prepared mind.
Pilots' information and analytical skills may be poor amid the confusing moments just after everything has clearly gone to heck. This is the strongest argument for accepting a forward + / - 90 degree target for an-off airport landing.
But if you're gonna turn around, this is what I figured out:
When the reverse course decision is made, job 1 is to reduce the vector velocity away from your intended landing spot. Turning is the only way to do this. The earlier and faster you turn, the greater the likelihood of reaching your goal.
By turning hard and early, you a) shorten the net distance, b) allow time to stabilize and assess the view, c) improve the odds for at least a successful off-airport landing nearer to those valuable emergency services on the field.
Slowing to anywhere near stall speed in a steep emergency turn is a bad practice in light aircraft and really bad in heavy ones.
The safer alternative is to crank in the 'maximum allowable' bank as soon as the decision is made, simultaneously pushing the nose down for acceleration. It is counterintuitive to dive when you only have x much precious altitude left, but doing so allows steepening the turn so that travel away from the airport can be stopped at the earliest possible time. If this is done in a smooth and 'aerodynamic' manner - flying through the turn rather than skidding - , then the total energy lost is not going to be vastly higher than optimum. (a la Bob Hoover) You will have traded some of the altitude for increased velocity, and the rest for the vector velocity cost of the turn.
Assuming your departure was straight out, the first 90 degrees of turn (into the crosswind) gets you to zero increase of distance from the airport. Ideally, this must be completed as soon as possible. The next 45 gets you onto a very nice converging downwind angle with your target, with time for shallow bank adjustments on the way in. When you are on the 45 and set up for max glide distance speed, you have a familiar, stable context and can then quickly intuit if it's going to work or if you need to find an alternative 'plan C'.