Are there any Hardware RAID Cards with Batteries that will work in a Mac Pro 3,1?
As the 3,1 is essentially an unsupported machine [sadly] I'd go for a second-hand Apple card from eBay etc, to save any possible compatibility issues with a 3rd party new card that may never have been designed with the 08 Mac Pro in mind.

The official Mac Pro RAID cards themselves come in 2 flavours, pre &amp; post 2009

Unless it's a refurb, I would also change the battery, new ones are still available.

UK prices, old RAID card for an Early 2008, approx £200, new battery £50

See Mac Pro (Mid 2012 and earlier): Frequently Asked Questions about the Mac Pro RAID Card and Xserve RAID Card