Not that it “couldn’t help”, but looking at it from a dollars and cents angle you’d be spending loads of YOU’RE money to resolve the issue, and it might not be practical at all. Shaft side isn’t just fastening sound board or carpet to the wall and away you go, it involves professional contractual advice, some sort of design and then application and that application to not interfere with shaftside equipment, AND meet fire codes. Not to mention anything inside the shaft requires a licensed elevator person be onsite to assist at every turn ($$$), and the lift is non operable as this type of work were to proceed. So you can see this is a big fat check to write to simply address your concerns.
Might be worth it to investigate a sound proofing professional to visit your property and see what he/she recommends understanding it’s not feasible to get inside the shaft. Determining “what” the noise is, “where” it’s migrating to and “how to deaden it” is their job.
I bet no one hears the noise like you do. By now your ears are set and tuned for it (snap…thunk…snap… thunk).