if my MacBook Pro battery model is A1321 would the A1281 battery work

I have a MacBook Pro mid 2010. works amazing I only upgraded the memory from 2 and 2 to 4 and 4 not to long after it started ghost clicking and switching screen

YouTube vid would pause and full screen then un full screen basically creating new folders opening files all sorts of things

I noticed it did this when it started getting hot. so I want to change the battery because I got the service battery message and I read in other forums that it the thermal paste for the over heating and then some say the battery Is swelling which can tamper with the track pad and cause ghost clicks…..

so I wanted to know since the model number on the battery is A1321 and the battery being sold here is A1281 would it work in my MacBook Pro?

OK, here’s the snapshots:

THIS THE COMPUTER ON AWAKE

THIS IS THE READING AFTER THE SIMPLE TASKS AND CHARGER IS PLUGGED IN

At this point I have to replace thermal paste to minimize over heating and I want to change battery to stop the ghost clicks because I do know the ghost clicks happen because of the battery swelling,

I can’t install much or update the OS because of the heating and ghost clicks but this happens when its overheats. the computer Doesn’t shut down or run slow so I can say it’s not the CPU.

I also notice on the image of the battery for sale its has the same model number as my battery but the item labeling on the site was where my curiosity was sparked.

About three days ago I cleaned the fans and compressed air cleaned it. The system gave me a full 36 hrs of no issues until it started again but not as bad as it once was.

Is TG PRO FREE or trial at least. The mid 2010 one looks like mine but I no the connector was the issue for most buyers of this product. Then the compatibility and specifications it kinda threw me off. Because my initial journey was to update the OS system hoping it would solve the ghost clicks and that process took me down a rabbit hole which lead me to understand even more that the battery thermal paste and over heating was and issue if I can solve these minor issues then I can clear the hard drive of unwanted stuff giving the computer more work space and then update the operating system and then progress with any other installs....

SORRY FOR THE LENGTHY EXPLANATION