I wouldn't expect that the rear wiper is related to the remote issue.

My remote didn't work when I bought the car some 25 months ago, the lock cylinder breaking (on a rather rainy night) was the impetus to fix both the cylinder and the remote system. I found a good used FZV module on Ebay for under $20. On my E53 X5, it's modestly bigger than a credit card, and at the price I paid - no concern to repair or even to diagnose it.

I repaired my door lock cylinder. Then I swapped out the (possible) bad FZV module with the used one I bought and BINGO - the key remote works. Problem solved.

One bit of advice, check your FZV module on your car to confirm the BMW part number and operating frequency. BMW uses different ones depending on where the car is sold (US, Europe, etc.) so as to use the legally allowed frequencies.