You don't need to do network sniffing, you just need to run a server that displays its connection parameters in detail. Contact that server from your client.
Some existing servers:
Qualys is best known for their server scanning but they also have a page that reports client information.
How's My SSL reports client information.
I don't think that app is using http at all. I've seen similar behaviour with WhatsApp in which https is only used at the registration phase and then it switches to noise pipes which is not intercepted by http proxies.
A workaround is to use mitm_relay. It listens raw TCP/UDP packets, encapsulates them in http and forwards it to http proxy (burp). The only ...