Your 316 grade wire has >800kg breaking load (BL) and the WW wire 861kg and >4.5 times the price as you say - cripes. WW don't quote a grade but trust/assume it is 316
www.whitworths.com.au/wire-1-16-1-x-19-s-sThe reel photo in your Better Hardware link seems to be of 7x19 or 7x7 wire. But the close up photo as below is 1x19 and seems to have a longer helical pitch as per normal 1x19 than the reel photo which I assume is what you were alluding to here
and the existing wire is wound less tightly
All these decisions carry some sort of risk but in this case seems like not much. Is the wire you have like the close up photo below or the reel photo? If you have >800kg BL wire with 316 grade the difference in 800kg and 861kg is not much obviously. Assume you have the hydraulic swage press needed?
www.miamistainless.com.au/3.2mm-w03.2119mtr