change the batteries with the game on and the settings/scores will not change.

I think you would need to have the game turned on while the batteries are changed.
I have remote battery packs so I'm comfortable doing it. With batteries on the board I would be a bit afraid of pushing or pulling on the MPU while it's powered up.
Guess it depends on how attached you are to that high score!

How about installing a remote pack while the old pack is still in?

Holy smokes, it's been almost 20 minutes without anyone mentioning NVRAM!

Edit: Seriously though, NVRAM is the way to go, I just haven't gotten around to it yet...

If you are soldering the remote pack in TURN IT OFF

If you are installing NVRAM --- also TURN IT OFF

To the OP, while going to NVRAM or a remote battery holder is a good idea, both will erase your high scores and settings. I'd opt to just replace the batteries with AA Lithium batteries with the game on. Just pay attention to the polarity as the (+) of all the batteries are going in the same direction.

He was only asking about changing batteries and saving settings, not alternatives. That said, while NVRAM is ideal, it is not for everyone because it requires desoldering and soldering chips to boards on older games like his. You need the right equipment for that, in addition to having very good soldering skills. There is no room for error and you can do a lot of damage if not done correctly. Instead the simplest (and cheapest) option for a novice is to simply use a remote battery holder.

Live life on the edge

At the Williams service school for the early systems (Gorgar time frame). We were told to replace the three batteries with the game powered on in Game Over mode.