I know this is a smidge old but I wanted to get a response in.
My solution is to begin the assault immediately after a highstorm by ripping the exterior of the wall off with a thunderclast or fannahn-im, then have metacha-im break through the aluminium layer with thrown boulders. Once this is done, there are two parts to the plan to actually take the fortress. The first is to use the (likely temporary) lack of aluminium to get makay-im (and potentially even nex-im if they can inside without being noticed) within the fortress. They are going to lie in wait for a short while, until immediately before the next highstorm. The second part is to attack with Magnified Ones and a large group of Regals/ordinary singers to force a ground response. In the ensuing clash, there should be a number of dead and mavset-im will take their place, pretending to have been wounded instead (specifically, they will take the place of ordinary soldiers who can't heal with Stormlight). They'll almost certainly be put in some sort of infirmary, and they will remain there pretending to be wounded until the next highstorm, the same as the makay-im in the walls. Once this preparation, I will continue with a series of skirmishes like the one I just described, but with the addition of shanay-im to draw out Windrunners. The objective of these skirmishes is to drain the enemy's Stormlight, and all the timing has been chosen so as the give the largest window of time to do this. Now, it is the time to strike. The makay-im will assassinate the stronger Radiants in their sleep and the mavset-im will sow chaos and confusion while there is another large attack, once again with the thunderclast and metacha-im.
With regards to forces and casualties, I'm thinking one thunderclast to be my wall-breaker and big thing that demands attention during the initial and final assaults, a dozen or so metacha-im to act as artillery, six to thirty-four makay-im (depending on what I'm able to be allocated; the first number is how many Radiants of the Fourth Ideal or higher there are in the fortress; the second is how many Radiants of the Third Ideal or higher there are), a handful of nex-im if any, five or so Magnified Ones so that I can switch them out and let them heal, fifty to seventy shanay-im to interfere with Windrunners during the skirmishes (I'm assuming that the duel dynamic is in place during this time period and that the Windrunners will be hesitant to let their squires die against experienced shanay-im), fifty or so mavset-im to masquerade as wounded soldiers (in the hope that I can kill a bunch of them during the first assault) and as many Regals (primarily direforms since stormforms will be worse than useless in the Purelake) and ordinary singers I can get so as to not have my siege broken. I'm estimating fairly low causalties for my Fused since the bulk of them will be in hiding for the duration of the plan (the makay-im will withdraw after the assassinations, though the mavset-im will likely die after inducing panic), and the others will be engaging primarily in skirmishes where they will withdraw after provoking a large response (out of these, the Magnified Ones are most at risk). The thunderclast's body may be destroyed but that isn't a tremendous concern. There will likely be a steady stream of casualties among my non-Fused troops the whole time, with casualties peaking on the first day (when they will need to engage in a ground battle with the ordinary enemy troops) and on the last day (when my forces will storm the fortress). I'm not sure what the precise numbers would be.
