My 05 bonneville pulls a bit when accelerating betwen 15 an 20 mph but only afte ive driven it for like 20 mins or more

Check compression. If its a 4 cylinder that's fairly easy with a compression gauge. All 4 cylinders should be around 180-190 psi. If all 4 are low suspect a jumped timing chain.

Resetting a CPU after a battery change is done with a code reader /scanner

check wheel alignment and wheel weights

maybe a map sensor, the large sensor inline with the air tube just aft of the airbox. it could also be a MAP sensor or manifold air pressure sensor. both are respnsible for correlating with the engine computer to control fuel.

Try cleaning the mass air flow meter. There;s a spray cleaner you can get at an auto parts store

Retrieve transmission faulty code first and check the transmission oil level.
Every time you turn off the ignition key the PCM act like that
The PCM will see the problems every time it happens and only in the same ignition key cycle.

hope it helps

You are correct on the struts and sway bar. Your car is FWD so it will always pull a little bit when accelerating and braking.

It sounds like you had a chunk of snow or ice on your wheel in the inside. This would throw off the balance terribly. When it finally came off the issue was gone. Nothing to worry about. its all good now.

check tires for balance or defect

my pal jojo had this problem just 3weeks ago and his fix was plugs and wires.hope this helps.