I might have misunderstood what you wanted to achieve in my previous reply. The following returns the DateTable[Period] where TimeRecords[Date] falls between DateTable[StartDate] and DateTable[EndDate]:
=CALCULATE (
MIN ( DateTable[Period] ),
FILTER (
DateTable,
TimeRecords[Date] >= DateTable[StartDate]
&& TimeRecords[Date] <= DateTable[EndDate]
)
)