The UPS in that Amazon link does not have a pure sine wave output.
The price point for these consumer units, in my opinion, is low, because these units are not designed to last longer than one or two sets of batteries.
They are also not engineered to operate for many hours continuously or to rack up many hours of operation during their lifetime. They are designed to operate only when the power goes out, and not robust enough to run frequently. Most of the time they pass the AC input through to the output and are not converting DC to AC until the power goes out.
UPS batteries are not built to be cycled many times, and UPS systems are designed to charge those batteries and will not have the proper profile or power to charge larger a larger AGM battery bank of deep cycle batteries.
While it may appear that someone can Gerry-rig components like this, what cannot be verified is the long term effects on an expensive set of AGM batteries being maintained by equipment that is ill-suited to the task, unreliable and potentially dangerous.