As far as I am aware, no Southern drivers are currently competent on MPVs, and only the tiniest handful are able to drive locomotives of any description (mainly a couple of drivers still competent on 73s for rescue reasons), therefore there should be no reason that railhead treatment trains loaded with de-icer should not run, except perhaps if they need to be piloted into any of the depots or sidings.
(Note that some other TOCs such as SWT do crew MPVs, or have done in the past.)
As for "ghost trains" formed of usual stock and designed to keep lines clear of ice by means of regular ECS running, these would be affected if they were run by Southern, but there is every chance that the signallers and the relevant control people will be able to devise coverage of the mainline by running GX and Thameslink over different selections of running lines, if need be. London Overground are also in a position to run ghost trains on the mainline from New Cross Gate to Norwood Junction if necessary.