Federal law allows each state + DC to appoint their apportioned electors any way they choose, and this right extends to the settling of disputes in the popular vote. The deadline for resolving these determinations, however, is six days prior to the meeting of the electors to cast their votes ("the first Monday after the second Wednesday in December").
Your assumption that there is a chief elections officer in each state who only votes in the case of a tie isn't accurate, and I don't know of any state that does so. Some states might hold a quick revote (if they even have time to do so after the recount deliberations and inevitable lawsuits have concluded), while others will likely have coin flip-type solutions. It all depends on state law.
Now, who wants to research 51 laws on how to break a tie in the statewide/districtwide popular vote?
