The standard battery is a very commonly used 12V/7Ah Sealed Lead Acid Cell. Verizon only provides the battery at installation. I suspect Verizon simply buys these batteries from whoever their current low bidder is. While you can order a replacement battery from Verizon, is is 2-3 times the 'going' rate.The battery has a nominal life of 5 years, however depending upon a number of factors, YMMV.
A replacement 12V/7AH battery can usually be found for less than $20, and you don't need a degree in rocket science to replace it. There is absolutely NOTHING special about the battery used in the Battery Backup unit. These batteries are even offered for sale on ebay saying they are for FiOS battery backup units.
My advice is that if you suspect a battery problem, it just isn't worth messing around with. If your landlord is like mine, the odds on her replacing anything that she doesn't absolutely have to are somewhere between slim and none. The battery is consider a consumable.
Just buy another battery, and replace the existing battery. As long as the terminals fit, it is 12V, and fits in the space, whether it is 7AH, 7.2AH or something else near 7Amp Hours, it will work just fine.