I'm fairly certain the AT-1 is centre weighted. I just tried mine and the metering is steady and progressive moving from ground to sky. It's unlikely you have the wrong battery, so I'd check electrical contacts. Also, try taking off the lens and see whether the meter still acts erratically. In the end, if the meter is indicating the correct exposure, I'd live with its mood swings.
The AT-1 is mistakenly overlooked, because it emerged in a period when automatic exposure was a must-have accessory. It wasn't even available on the Japanese domestic market, but was aimed at emerging countries where the extra price of auto was a deal breaker. In every other way it's a standard A-series camera. It has some rare assets for a full manual camera today, it takes a readily available battery (not an arcane mercury cell), it's fairly light (unlike most full manual cameras that are heavy metal), it has a proper PC socket for flash, a stop down switch and that huge shutter speed dial under the wind-on lever. The AT-1 isn't as 'cool' as some classic manual exposure cameras, but it is extremely practical and they can be picked up very cheaply. They were often pampered and are likely to be in much better condition than your typical Nikkormat/ Spotmatic. Just remember to turn off the meter when you're done, like an OM-1.