This is what Tesco say about their policy:
"Recycling your batteries
Tesco is part of a compliance scheme and offers battery-recycling points at Express, Metro, Superstore, and Extra stores. In 2020, 566.57 tons of batteries were collected in Tesco stores.
You can recycle all household batteries at our recycling points, including AAA, AA, C , D, 9 volt, button and those from appliances such as mobile phones, laptops, hearing aids, watches, cameras, cordless power tools, electric toothbrushes, razors, toys, burglar alarms, and hand-held vacuum cleaners.
Lead acid batteries from cars are not suitable for instore recycling points."
Our approach to managing waste from our UK operations
Unless your Tesco is not an Express, Metro, Superstore or Extra store*, you should ask the manager why they aren't following Tesco's own published policy
'* I don't have a local Tesco so no idea if there is another type of Tesco store, but 5 seems like plenty to me