In the long run, sink rate is not due to a "shortage of lift" but rather to a shortage of thrust.1 If the angle-of-attack were held constant, then in the very short run, increasing the bank drives a brief shortage of (the vertical component of) lift-- this is involved with the dynamics of causing the flight path to curve down as the airplane accelerates (gains airspeed)-- but that's not what you need to be focusing on. After a very few seconds, the aircraft will have gained enough airspeed to restore the vertical component of lift to essentially its "normal" value. So... it's complicated.
Furthermore, consider this-- in your problem, the specified airspeed is assumed to be fixed. This means that the angle-of-attack is certainly not fixed. As the pilot enters the turn, to hold the airspeed constant, he must be increasing the angle-of-attack sufficiently that the vertical component of lift stays (very nearly) constant throughout the turn entry. So there's no shortage of the (vertical component of) lift to speak of-- not even temporarily, as the aircraft is entering the turn.
So the steeper bank turns are being flown at a higher angle-of-attack than the shallower turns. But that's never going to offset the effects of the steeper bank angle enough to actually cause a decrease in sink rate. The key factor is rather that you complete the turn faster when flying at a steeper bank angle, so the altitude loss per degree of turn is decreased.
Nonetheless, the fact that the steeper bank turns are being flown at a higher angle-of-attack does bias the results for minimum altitude loss per degree of turn toward a somewhat higher bank angle than we'd obtain if we specified that the angle-of-attack were held constant (and the steeper bank turns were therefore being flown at a higher airspeed than the shallower turns)--at least so long as we remain on the "front side" of the drag curve.2
Actually I initially missed the part about the specified airspeed being the exactly the same as the wings-level best glide speed. So we're not deep on the "back side" of the drag curve, but we're not way on the "front side" either. Rather, we are (in the wings-level case) right at the very bottom of the drag curve (i.e. drag is at a minimum).
A key point is that when you are turning, you don't care how far forward you go, you just want to minimize sink rate and maximize turn rate, so there's no reason to fly at best glide speed, you instead want to fly at the min sink rate speed for that bank angle, or arguably even a bit slower, to maximize the turn rate, and therefore minimize the altitude lost per degree of turn.
So starting from the wings-level best L/D (max glide) speed and angle-of-attack, increasing the angle-of-attack as we enter a turn is initially beneficial, in terms of minimizing the altitude lost per degree of turn.
At some point, however, if we continue to increase the bank angle while holding the airspeed constant, we will end up increasing the angle-of-attack so much that we are mushing along near the stall at such a high angle-of-attack that the sink rate, and therefore the altitude lost per circle, will be severely increased. (In fact, if the bank angle becomes steep enough in relation to the airspeed, we actually reach the stall angle-of-attack if we attempt to hold the airspeed constant by continuing to move the stick or yoke aft-- see this related answer to the present question.)
It seems odd that the aircraft has apparently not yet reached that unfavorable part of the flight envelope when the bank angle has been increased all the way to 60 degrees, with no increase in airspeed beyond the wings-level best glide angle (best L/D) airspeed. For sailplanes and light airplanes, I've typically seen bank angles around 45 degrees recommended to complete a turn with minimum altitude loss, while flying (as best as I recall) at the wings-level best-glide airspeed.
Perhaps there is an unusually wide spread between the wings-level airspeeds (and angles-of-attack) for max glide (best L/D), and for min sink rate, in this aircraft. Does the manual publish the airspeed for minimum sink rate in wings-level flight, for the same aircraft weight at which the wings-level best glide speed is 125 knots?
Also, consider this-- if we are free to choose any bank angle, is keeping the airspeed right at the wings-level best glide speed really always the very best strategy for minimizing the altitude lost per degree of turn, or at least a close approximation of such? Enquiring minds want to know...
Yes, as has been covered elsewhere on this site, the geometry of the force vector triangle is such that (the vertical component of) lift must be slightly less than weight whenever the flight path is descending (or climbing!) in a steady-state (constant-airspeed) condition, but it's a misconception to think that this discrepancy is what is causing the flight path to descend (or to climb!). Rather, this discrepancy simply very slightly decreases the airspeed at which the aircraft will be in equilibrium (i.e. the airspeed will be constant), for any given angle-of-attack and bank angle and glide angle (or climb angle!). In other words this dynamic is very much a sideshow, not the main event.
If the aircraft were initially being flown in slow, "mushing" flight, on the "back side" of the drag curve, then any further increase in angle-of-attack would clearly be detrimental, and if we specified that the airspeed were held constant, we would bias the results toward a lower optimum bank angle for minimum altitude loss per degree of turn, than if we we specified that the angle-of-attack were held constant. Exactly where on the drag curve we'd need to (initially) be flying so that we'd end up computing the same optimal bank angle to minimize altitude loss per degree of turn regardless of whether we held airspeed constant or held angle-of-attack constant is not completely obvious (and perhaps grounds for an interesting exercise or new ASE question.). It appears that this "break-even" point would correspond to a starting airspeed somewhat slower than the wings-level max-glide (best L/D) airspeed, but well above the wings-level min sink rate airspeed.