Вы не можете просто поместить @receivable, @invoiced и @receivable + @invoiced в один оператор выбора. (Они не будут хранить значение по порядку. Они будут выполнены одновременно.)
Сначала вам нужно сохранить значения в @receivable, @invoiced, а затем использовать подзапрос для вычисления суммы:
set @start_at = '2019-01-01';
set @end_at = '2019-01-16';
set @receivable = 0;
set @invoiced = 0;
SELECT *, A.[Account Receivable] + A.[Invoiced] AS TOTAL FROM (
SELECT DISTINCT Customer.custnr 'Customer Number',
Address.name 'Name',
@receivable := sum(case
WHEN [condition1 <= @start_at]
AND Transactions.`key` not in [subquery]
THEN Transactions.amount
ELSE 0 END) 'Account Receivable',
@invoiced := sum(case
WHEN [condition1 between @start_at and @end_at]
AND [condition2]
AND [condition3]
AND Transactions.`key` not in [subquery]
THEN Transactions.amount
ELSE 0 END) 'Invoiced'
FROM LocalCust
INNER JOIN Customer
on Customer.`key` = LocalCust.customerkey
INNER JOIN Address
on Address.`key` = Customer.addresskey
INNER JOIN Location
on Location.`key` = LocalCust.localkey
INNER JOIN Transactions
on Transactions.localcustkey = LocalCust.`Key`
GROUP BY Transactions.localcustkey) A;