I'm going to be making a similar cake for my daughter's wedding in October using the same size square tiers as you.
After taking a class this week with a very experienced cake decorator in my area, one of the recommendations she gave me was to do square styro risers in between and transport unassembled. I may assemble the bottom 2 tiers for transport and then finish off the other 2 at the venue.
I am making sugar roses for between the tiers, so she suggested wiring them right into the risers, then carefully box them up separately with lots of padding. (making sure to take some extra flowers,,,just in case!)
I use bubble tea straws for support, so that is the main reason for not doing complete assembly before transporting. My oldest daughter was married last month and I did 10" square with rounds of 9", 7" and 5" and using the straws and same method of delivery outlined above, the cake was fineI (It is pictured in my photos) It was my first wedding cake and I was very nervous, but I was quite pleased with how it all came out.
I'm sorry I am not very familiar with SPS, so possibly someone else might be able to give you advice with that if you prefer to use that method.
Good luck with your cake!