Suppose I make a call (Samsung A03), to a system I know is going to use keypad prompts.

I put the call on speaker, then call up the keypad.

While I'm waiting, the screen fades out. If the list of me u options takes a while for the robot to read, the screen fades out. You get the picture.

How do I go about increasing that delay? Is it possible?

Settings> Display> Screen Timeout.
Typical is 2 or 3 minutes.
Set to maximum, which is 30 minutes on my Android phone.
Don't forget to reset back to a reasonable Screen Timeout after your call is complete
or your battery life will suffer due to the 30 minute settings change.

I found that setting. It works great for any app other than the phone.

When using the phone, the screen still dims after just a few seconds.