Can a Military-Grade wipe harm an SSD?

I've written an answer in the past on wiping drives. Its tangential to your question but there's no practical purpose in doing a 35 pass DOD wipe on a modern drive. The guttman method, which the DOD wipe is based on assumes you were wiping a drive with much bigger data domains than a modern drive, with unknown internal encoding.

As for damage, while 35 passes is quite a lot of writing, modern SSDs, even MLC and TLC drives quite happily handle thousands of cycles and terabytes of writes -There are many endurance tests floating around online such as this.While entirely pointless, this isn't likely to appreciatably reduce your drive's lifespan. What you really want to do is a sata secure erase - which tells the drive you want those sectors erased, and incidentally dosen't need 35 passes.