Actually the services would give a 'shiny shit'. Sorry to disagree.
If baby had a reaction to the pen or whatever products were used to remove it (e.g. skin rash or ecczma, or an allergy triggered) and baby was presented to GP, HV (or a nurse in outpatients) (which you would do as responsible parents) and stated it was as a result of DH drawing pictures on his 5 week old newborn with a sharpie pen for his own amusement when drunk, then it would get flagged. For reasons several PPs have noted, including DH's attitude to his baby and that he did it whilst 'in drink' .
That in itself might not result in a full fledged child safeguarding enquiry, but it'd certainly be logged and depending on which LA or CSD team manager saw it, possibly also followed by an unimpressed telephone call from duty social worker.
But it didn't and hasn't (caused harm), and OP has nipped it in the bud with DH, who has realised it wasn't ok, and has promised no repeat.
I think OP's shocked & protective reaction for their baby was spot on, no harm luckily occured, she dealt with it quickly, and her DH realises it was inappropriate.
But I am surprised at the minority of PP who have been dismissive about OPs concern at this moment of poor judgement exercised by her DH.