If you think you can manage the train after arrival (Thameslink, hourly only after 11.30pm), I'd suggest train to Farringdon, a night in the excellent Premier Inn Farringdon Smithfield (maybe with a pint at one of the Smithfied market pubs that open early for the market porters), and then Elizabeth line to LHR the next day from Farringdon.

Farringdon is OK at night, especially during the week.

The Premier Inn has family rooms around GBP 180-200 non-refundable if you book now, kids under 16 get free breakfast (two children per one paying adult).
On Thameslink, you can add children under 16 to your ticket for GBP 2.00 each, so an anytime single from Gatwick to Farringdon is currently 18.40 for one adult and two under 16s.