You are trying to generate a DTMF signal during a call, not possible without root... from what I can determine.
However, if you could record this tone accurately (since it is the same defined dual tone multi-frequency generated every time), and if reproduced in a manner confirming to tone and pulse recognition by the receiving device (should be national standard), and if that were streamed through the call of the device (only possible with Sony, I believe)... then in theory that should work.
However, to my knowledge only Sony devices can send audio generated through a call, instead of just over it.
There is the SMALL chance the recognition equipment is antiquated enough where blasting this reproduced tone as a media file loud enough to be picked up by the microphone and sent over the call could result in desired action. But I would venture to guess that it is HIGHLY unlikely the equipment is over 40 years old. Otherwise, a Sony device would probably be the only device that MIGHT achieve this without root.
Manufacturers are weird about certain things like that.
Interesting,
What you're thinking is what I was thinking. The option exists, only works on Sony.