Calculate % increase from last Period (Month)

You've got the variance caluclation correct, you just need to actually calculate the previous month.

Actual LCR = CALCULATE(
   AVERAGE('Resource Actual'[Quantity])
   ,'Resource Actual'[Category] = "Cost Rate"
)

Actual LCR LM =
VAR last_date = LASTDATE(ALL('Resource Actual'[Date Column]))
VAR current_date = LASTNONBLANK(DateDimension[Date],[Actual LCR])
VAR ly_date = NEXTDAY(PREVIOUSMONTH(current_date))
VAR date_context =
    DATESBETWEEN(
        DateDimension[Date]
        ,NEXTDAY(PREVIOUSMONTH(current_date))
        ,LASTDATE(current_date)
    )
VAR result = CALCULATE([Actual LCR],DATEADD(date_context,-1,MONTH))
RETURN
result

Variance = DIVIDE(
   [Actual LCR] - [Actual LCR LM]
   ,[Actual LCR LM]
) * SIGN([Actual LCR LM])

This should get you what you want if not let me know.