However, every time i reboot the router the DOT protocol stops working (though it remains enabled in web ui) but instead of the cloudflare's DNS my ISP's DNS shows on https://www.dnsleaktest.com/
Can't reproduce that here, works for me right away on a reboot. Make sure your NTP server is properly configured, and that you don't have any custom iptables/dnsmasq configuration that conflicts with DoT. You also might want to set DNS Privacy mode to Strict if it wasn't.
That transparent proxy probably just intercepts traffic on port 53. DNS over TLS avoids that by using port 853 instead. Until your ISP decides to block/redirect that port as well...
If possible, I would consider looking for another ISP who doesn't try so hard to monitor ALL of your traffic.