The method I'd suggest you use is tSQLt.ResultSetFilter(). This takes a parameter for number of the result set to return and calls your code under test (StoreProcName in your example), returning that result set, which you can then use Insert..Exec to capture.
The down side of this procedure is that it only captures that one result set per run - so you need to call it multiple times to return all of the result sets. I usually only look at one result set per test, allowing me to concentrate on answering one question in that test, but if your result sets interrelate and you need both to return for your test to be evaluated, then you will need to call tSQLt.ResultSetFilter and hence the code under test more than once in your test (the manual has more info on this situation)
As an aside, I have previously blogged about some unexpected behaviour I encountered when using insert..exec with SPs that return multiple identical result sets which may be of interest.