The column data types in the flat file source should match the destination table. So the flat file source column should be changed to DT_R4, or the database column should be changed to FLOAT.

If the input file really does have a value that doesn't fit a REAL (DT_R4), you'll get an error from the flat file source component rather than the destination.

I wouldn't rule out something like a stray comma in the input file that is throwing it off. Try copying the error-generating line to a separate file, then run it through your package in BIDS with a data viewer on the output of the flat file source. Verify that the column in question has the value you think it should.