I'm not familiar enough with localtunnel to tell you what's wrong there, but I can tell you how to accomplish your same goal using Telebit:
(p.s. Did you figure this out? If so, I'd love to hear how you did it and I'm sure others would too)
Install
curl https://get.telebit.io | bash
You can also install via npm... but that isn't the preferred install method at this time. There may be some caveats.
The random domain you get is attached to your account (hence the need for email) and it's encrypted end-to-end with Greenlock via Let's Encrypt.
Configure
./telebit http 4000 xyz
The general format is
./telebit <protocol> <port> [subdomain]
It's not just https, you can use it to tunnel anything over tls/ssl (plain tcp, ssh, openvpn, etc).
Custom domains are not yet a generally available feature, but they're on the horizon.