Should I change the battery at the authorized store or local one?

Hello there. I live outside the US and there’s no Apple store in my country. My 2 years old iPhone XS Max battery starts deteriorating and the battery health is at 81%. It overheats easily because of the battery. I called the Apple authorized service provider and it costs 129$ for the battery and 35$ for the labour, which is much much more expensive than the reliable local repair shop which charges only 60$ for the battery.

Is the quality the same as the original one?