The early entry tours we took ended at the Basilica. We only had small bags (basically a purse or messenger bag) so did not check anything before going through the museums. You can also take a purse or messenger bag through the Scavi, but not any sort of back pack.
If you have a back pack, you check it to the right of the stairs that lead up to the Basilica (as you face the church). So if your tour ends, like ours, in the Basilica, go down the stairs on the left as you exit and look for the security line and bag check. My recollection is that you have to go through security to have the bag scanned before it is checked.
The line can be long, which concerns me since your Scavi tour is at 11:30. Are you certain the museums tour will end in time for you to get to the Scavi tour, especially if you have to check the day pack? Can you manage with a purse or messenger bag so you do not have to check them before the Scavi tour?