You want to sum a portion of one row of an array, where the date is less than or equal to a specified value.
First, let's figure out how to get one row of the array. The INDEX() function
INDEX(array, row_num, [col_num])
will return a whole row if the col_num is set to zero. So this function
returns the row of your data where the Account(?) is 403300. You can check this by highlighting the formula in the formula bar and typing F9. That will show the value of the formula - an array of the data in the 403300 row.
Now you just need to add up the portion of that row where the month is less than or equal to the specified month.
SUMIF() will do this.
SUMIF() checks a specified range (your dates) matching a criteria (<= your specified month) and sums the corresponding cells in the sum_range (the row chosen with the INDEX() formula above). Putting this all together, and using the mocked-up data table below, this formula
in G4 gives the sum of the account in D4 through the date in E4.
I've put everything on one worksheet and without dropdowns, but you can easily add these features. If you really need to specify the worksheet with a dropdown, you do have to use a lot of INDIRECT()'s, which gets a bit messy. I came up with this, where the sheet name is in C4:
I hope this helps, and good luck.