How many amperes are in 1250VA?
The adequacy of your UPS as a charger depends upon the type of battery for which the charger was designed. Most UPS's are made to charge gelled electrolyte batteries. If this is the case, you may find that it takes longer to charge your wet chemistry battery or that the battery never fully charges. But it most likely will not damage your battery as long as everything is a 12 volt system.

There is no problem running your fan when the battery is charging. It may take a little while longer to charge the battery as a result but nothing will get damaged in the process.