I think this is an issue with the design of the Stratus devices. They seem to not do a good job of handling a strong signal that is nearby, that is the AGC can't compensate for the short distances and strong signal. I have also seen what I call the "behind me syndrome" where the indicated position falls further and further behind the true position. Remember the SkyBeacon is only 10 to 20 feet from the Stratus and is broadcasting once every second. The ADS-B Out from your SkyBeacon is being received at the Stratus. The broadcast is totally independent of the FAA system and the Stratus, so if the ground stations see your broadcasts showing where the aircraft is located, so should the Stratus when it receives the signal from 20 feet away. I have seen Stratus that won't receive a nearby ADS-B Out signal until you physically move the Stratus to more than 50 feet from the airplane. I think it is a design deficiency and possibly a software bug in the Stratus.
If you don't tell the EFB software what the ICAO ID is of the own ship, then an algorithm needs to be able to detect the SkyBeacon that remains near your GPS position and altitude and then once it makes that determination, it should suppress the display of that ADS-B source as being a target and in fact is flying in near formation with you because it is you. Otherwise it will appear as a ghost target that is not real.