Short answer is mixing aerospike and solid fuel is probably not particularly useful
The possible advantage of the mass and complexity of an aerospike is efficient operation from sea level to vacuum, useful in a hyperthetical single stage to orbit or space plane design. It is also normally assumed the complexity/cost is recovered by making them re-usable.
Solid rockets cannot single stage to orbit in any sensible design due the poor performance (specifically mass fraction) due to entire casing needing to sustain chamber pressure. Solids work well as first stage boosters where high thrust and simplicity are useful, and they will be burned out and dumped well before making orbit (working better with rocket equations).
Achieving a steady acceleration from a solid is also tricky as mass burns off. Shaping the thrust profile is possible with careful grain design but getting a 'throttle bucket' profile would be tricky.
So we have a nozzle design really only useful for re-usable engines burning from surface all the way to orbit (and back) and a fuel system best used expendably for short durations as a first stage.
As noted in in question, cooling is a problem for aerospikes/variable geometry nozzles due to the shapes and need to be adjustable - cooling is normally achieved through one of the liquid propellants being routed through the nozzle, without them a solid fuel aerospike potentially ends up needing a liquid cooling tank system. Most solid rocket motors use ablative cooling for the nozzle which largely precludes moving parts.
This does mean that if ultimate performance was required from a solid rocket it might be possible to customize the ablator mix down the length of the nozzle to achieve a geometry change during the burn. This would most likely be heavier than a normal design, and exact ablator structure will need to be customised for each mission (as the nozzle widening rate will be fixed by chemistry, not a mechanical adjustment). This custom handling of ablator starts to reduce the simplicity, reliability and predictability of solid rockets that makes them useful in rockets.