I don't know if you can use MQTT as I don't think BI can subscribe to a MQTT topic. You can however use HA and the shell command integration to run a curl command to change a profile. Then you can setup a profile in BI that doesn't have any triggers enabled.
I have 2 schedules and 4 profiles, day and night profiles for both home and away schedules. When the home schedule is running I don't get alerts. When I leave home, HA knows this via presence detection and it runs a shell command to change the schedule in BI to away so I can get phone alerts.
So if you have weather setup in HA, you can use that to trigger an automation to run the shell command when there is a storm, rain or whatever weather condition you want occurs.