iOS shortcuts: Is there any way to input key presses into a call after it is connected?
FYI, I ran this and discovered that unless you turn on the speaker phone, the spoken audio will come through the phone headset.
You could try doing this as a shortcut.
Step 1, you would have to have the phone dial the sequence that you have listed with the pauses.  That would have to be a pre-set contact.
Step 2, you have to set a Wait. You'd have to set the seconds that you need to wait for the call to go through, step through the dial in prompts, before this next step.
Step 3, turn up your volume to 100%
Step 4 would be to speak your name. You enter in your name and you can choose the voice you want to speak it. (see top.)
Good luck!