If its not covered under the 5 year warranty, then buy a known make brand . Just make sure its the correct type ,especially if the car is fitted with stop/start system.

Not sure i'd class replacing a failing battery as 'servicing'. If the battery failed completely, somewhere far away from the nearest dealership, would they really expect you to take the car to a dealer rather than get a battery from a local garage/supplier?. I can't say 100%, but i find that unlikely, so unless someone knows otherwise, i'd say just replace it yourself. As long as the specs of the battery are the same, i can't see there being an issue.

Of course you could always get in touch with Hyundai customer services to be sure.

My i10 is 7 years old and still on the original battery, I was doing a 1000 miles a month in it till may, as others have said you might not be using the car enough to keep the battery in a healthy state.

As already said, it’s not a worry from the warranty point of view as long as it the correct type for the car. I’d buy one myself and fit it but take a look before doing so at the terminals as some are quite awkward and have a load of fragile electronics attached like our wee Polo TDi which won’t take kindly to heavy handed clumsiness. My swearing didn’t damage it thankfully though.