I'm assuming your problem is one way, you can't hear your child but you can speak loud enough for him/her to understand you, so I'm pretty sure this is a hearing aid amplifier use case. There are some with bone conduction headset and relocatable/aim-able microphone.
You could place the microphone on your back, close enough to pick your child's voice, so the apparatus is fully on your body, diminishing the need for the child to wear an uncomfortable device.
Just a few minutes of search turned out some interesting results, I used the following search terms: "hearing aid headset with microphone". And this is the one that caught my attention: https://www.amazon.com/Bone-Conduction-Hearing-Aid-Seniors/dp/B08M9DQ2JG/ref=sr_1_12?crid=OUEANOJ22IQM&keywords=hearing+aid+headset+with+microphone&qid=1636640208&sprefix=hearing+aid+microphone+and+head%2Caps%2C269&sr=8-12
I'm not trying to give product recommendation as I've never been a user of these equipment nor work for any related company.
I once knew a person with hearing problems, who was always wearing headphones. Later I found that this person had programmed something for using the smartphone as hearing aid. I do not know how this can be done, but I'm also sure that by now there are multitude of available apps for that. For this, I've seen wired microphone for smartphone with regular headphone output jack, so you could fix the microphone on the back of your jacket, route the cable to a designated pocket where you put the phone, and do the same with your preferred headphone. Search for "lavalier microphone with headphone output"
I´ve never tried bone conduction headphones, and I do not endorse use of headphones while cycling on routes with automotive traffic or the like, but I have used regular over the ear headphones worn not directly over the ear, but close enough to hear the music. This can also block wind noise to the ear.
I think a fully wired solution can be more practical, as will not require to keep multiple batteries charged, except for the phone, which most of us already do, and the full setup can be made "wearable" so it takes no more time than putting on a coat.
I have also used a bluetooth speaker inside the top of a backpack for riding in the city, The speaker volume is set close to a normal conversation loudness, so when there are no other vehicles close I can clearly hear the music, but just the rolling noise of car tires is enough to overcome the speaker, so the risk of not hearing an approaching car, siren, etc. is really low. I later found that this concept of using an open air speaker near the ears at low volume, does exist commercially: "neck speaker". The idea would be to make your child's voice louder. Again, you could combine this speaker with a suitable hearing aid or microphone amplifier.
Related anecdote: When I was a kid I had a small handheld voice recorder (the ones that used micro cassettes). The thing had headphone output and I really liked to use it to better hear faint noises, or as a rather poor electric stethoscope by pressing the recorder's mic into my chest. If by any chance you already had access to a similar device, you could try the concept before investing in more specific accessories.