Which format are you using to export to the other application?
Is the other application flexible enough to accept more than one format? You have more control in text formats for example and you can multiple process to include them.
That said for a single value it might be more practical to simple edit the value after export depending on the frequency of use, etc., etc.
Does the target application's data ingestion process have any settings that would allow blanks to be interpreted as zero? (I assume not bit there may be functionality that has not previously been revealed to you.)
Is your 0/0 calculation the result of part of the measure formula in the Summary? (I read you description and assumed it is.)
For reference here is a very old forum post for a suggested approach but it is based on fixing the challenge in the Table not the summary. Ignore the apparent posting date - it's an import from a much older forum. And, sadly, the links will no longer work even if you manage to figure out what they are supposed to be.
If the nulls are generated in the table and your calculation is, for example, Value1/Value2, then change it to If(IsNull(Value1/Value2),0,Value1/Value2).
If you are using a calculation if the summary itself, then I'm afraid you don't have the IF() function available.
If you want to ignore the error lines, you can "hide" the error lines from the summary by using the Matching values on the key field. But if you need the error lines, I'm afraid it would need a second pass in Monarch or Excel to correct the summary values.