OK, it's most likely not the regulator, and if it is, both the propane and butane regulators will freeze as it is just the diaphram that is freezing.
No, it's not safe to pour a kettle of boiling water over it, but possibly wrapping it with an old jacket may help
The real issue here is the nature of the gas, remember it is supplied as a liquid under moderate pressure, as we relieve the liquid boils to a gas, butane boiling pint is -2c, so when it's this cold the boiling action stops and no gas is produced.
Now the boiling point of propane is -42c so even in the harshest of weather we have seen it will still produce gas.
Butane is primarily a 'holiday' gas used for bbqs and summer pursuits, and maybe used indoors in the winter for such things as mobile space heaters, propane under current regulations is not allowed to be stored inside (and does not require it...so far!..brr)
So a changover for you sounds like the best move, do not attempt to do it yourself, get a qualified professional as appliance must be checked for compatibility (may need adjustment or replacement jets) and when the reg is changed and fittings have been made/remade a proper soundness test will be required to confirm there is no leaks!
I recently can across a diy cooker installed with a 'huge' leak under the appliance, how there wasn't an explosion is just shear luck, for the cost it's not worth it