[mTLS] Custom widgets do not work

### Preflight checks

- [x] I am using the experimental mTLS feature
- [x] I have reproduced this issue at least once after restarting the app

### App platform

iOS

### Device and app information

Model Name: Iphone 13 mini
OS Version: 26.4.2
App Version: 2026.4.1 (2026.1904)

### Home Assistant environment

Home Assistant Core Version: 2026.4.4
Installation Type: Container
Host 0S: Ugos Pro (Debian-based Linux distro for NAS)
Reverse Proxy: none
TLS Termination Point: no TLS for Internal url, Cloudflare Tunnel for External url

### mTLS setup details

Client certificate source: generated by Cloudflare
Client certificate format: PKCS#12
Certificate validity window: 6 April 2026 - 3 April 2036
Signing CA type: public
Server certificate issuer: Cloudflare
Full chain served by server?: unknown

### Failure symptoms

When does it fail?: on click
Frequency: always
Time of most recent failure: 2026-04-26 4:39:16.608 PM UTC

### Steps to reproduce

1. Click the lamp toggle widget or another custom widget

### Expected vs actual behavior

Expected:
The lamp turns on or off.

Actual:
The lamp does not turn on or off using both internal or external url.

Removing server and adding it again didn't help. I also noticed that WebSocket status remains 'connecting' all the time.

If the mTLS certificate is removed, the widgets start working again on the internal URL.

The problem is reproduced on two different devices.

### Additional context

[HomeAssistant-Extensions-Widgets.txt](https://github.com/user-attachments/files/27101822/HomeAssistant-Extensions-Widgets.txt)