There's a lot of missing detail here that will likely prove this is not a compliant installation. The problem with just the basic (or less than basic) details you have provided thus far is there is little specific requirements as to the rating and connection of transfer switches.
However, I'll first assume this is an Optional Standby System (Article 702) because it has the most lax requirements of Articles 700, 701, and 702. I'll first point you to 702.5 and quote the first sentence...
Transfer equipment shall be suitable for the intended use and designed and installed so as to prevent the inadvertent interconnection of normal and alternate sources of supply in any operation of the transfer equipment.
IMO, this would mean that the three (3) ATS' must be mechanically interlocked such that they can only connect to the same source concurrently.
Another potential violation is that I doubt you can maintain the ATS listing and land 1200A worth of conductor on each ATS line connection. If you use a lesser rated conductor scenario, you will be into tap rules, and will have to provide tap conductor OCPD before the switches... and then if you do that, you'll not be able to combine on the load side because then you'd violate the parallel ocpd prohibition.