Using EXISTS within a GROUP BY clause

You could emulate this behavior by counting the distinct classifications that answer this condition (per group):

SELECT 
    EOMONTH
    ,AccountNumber
    ,CASE COUNT(DISTINCT CASE WHEN classification_name IN ('cat1', 'cat2', 'cat3') THEN classification_name END) 
          WHEN 3 THEN 1 
          ELSE 0 
     END
    ,SUM(Value) AS totalSpend
FROM @tran_TABLE
GROUP BY 
    EOMONTH
    ,AccountNumber