Personally, I don't think it is at all against UC to monitor baby during a birth. For my birth, I only checked heart rate once. The rest of the time, she was kicking and moving strongly, so I knew she was fine. If I don't feel much movement, I'd check.
It's not that it's an emergency, but cords do get compressed sometimes and not feeling any movement at all isn't normal, so it might help actually ease any worries you might have if you check. Lowered heart rate during contractions and pushing is completely normal.
I did not use any U/S on my baby and especially not doppler because you are right, it is worse. It uses continuous high-frequency waves where regular U/S uses short intermittent bursts.
I used an electronic stethoscope (the BebeSounds) which has no waves at all. It's a regular stethoscope, it just has an electric volume control so the mother can turn the volume up or down to hear better. I found it quite reliable and a stethoscope would be as well.
Also, if you cannot find the baby's heartbeat while listening, it doesn't mean that the baby's heart is not beating. It's usually very easy to hear the placenta. If the placenta is whooshing, then the baby's heart is beating. It's the baby's heart that pumps the blood through the placenta. The only thing you have to watch out for is that you are not picking up your own heartbeat/vein pulses which are about 1/2 as fast as the baby's normal rate. To tell the difference, keep a finger on your pulse while you listen and make sure that the beats are not matching up.