So this means that the inverter heats the electrodes during operation.
Does not have to. If the lamp goe into the hot electrode mode, then really low power suffice to keep it there.
Yes, the cathodes are degrading a bit faster than at normal power, but on the emergency unit the overall burned hours are very low, so even when the lamp reach the lifetime about 1000 hours, we are talking about decades (in years) of the useful life. And for that ~40W power is sufficient without the external heating.
The problem usually arise when the lamp does not transfer to the hot electrode mode, then the lamp life become even 10 hours and below and that become the real lifetime problem.
And that is, what happen with many inverters: The low ballast power capability into a high voltage drop across the cold cathode operated lamp does not provide enough electrode heat to transfer the lamp into the hot electrode mode and then the life really counts in hours.
And there are circuit tricks to heat up the electrodes when the discharge is not there or the voltage drop is high (no discharge or cold electrode mode), yet do not consume any power on heating when the arc voltage become low (in the hot electrode mode), all that without any active switching element (the trick is, than the filament is in a diagonal of a bridge formed by secondary taps and some capacitors, what is highly unbalanced, so feed the filament with quite high power when the lamp have high drop, but well balanced, so no filament power when the lamp is in the low drop, hot electrode mode).