Unless a DOS driver was ever created for it in the past which has gone missing due to its age, its very likely that the driver was only ever made for Windows 3.x and maybe 95. I dont know of any wrapper program to get Windows drivers to work within DOS only. You might be forced to construct your own drivers.
During the period that that laptop was made controls in DOS were mainly Joystick or Gamepad through Game Port
https://en.wikipedia.org/wiki/Game_portOne option might be to find a Mouse or Track Ball that is external to laptop that connects through 9-pin serial that has DOS Drivers for it readily available.
In my travels through that time period there were very few applications that ran DOS only environment that gave Mouse/Track Ball functionality. Almost all DOS software was keyboard driven controls.
If you have Windows on it running and run the program while Windows is running and you have drivers for the track ball for Windows working, you might be able to run a program in DOS that has mouse/track ball control options and have Windows translate the controls to the DOS environment but I am thinking that sort of stuff with Windows/DOS co-existent functionality for the Mouse/Track Ball didnt happen until 95 or 98, and prior controls were strictly devices intended for DOS only environment support. Additionally games or software in DOS that had control options other than keyboard controls usually required manual configuration of non-plug-n-play settings where your telling it everything it needs to function as was sound cards that I had that had the Game Port option, as well as if it was a 9-pin serial device there was a serial configuration for it such as which com port the device is at and all that because there was no auto sensing like USB unless a DOS program checked for device handshake at COM1 and then COM 2 and all the way through say COM 4 and then time out with no device found if a handshake to device not found which I can see being made possible but never seen a software that did that that was DOS only.