Isn't the I9300 the 1gbp ram version?
Anyway, if certified in download mode you could just flash t999 firmware.
If that doesn't sound safe you can also put a t999 recovery on it and try doing a backup and installing a t999 rom
Sent from my SGH-T999 using Tapatalk
Sounds kinda like some of the "frankenphones" we've heard about over the last several months. Or it could also be a fake/chinese knock off. If I were you, I would see if there were any way to send it back to whoever sold it. If you plan to keep it though, understand anything you do is AT YOUR OWN RISK! The only absolutely safe course of action is to do nothing to it. If you try to flash it via Odin, and it allows it to flash, but you chose the wrong firmware, it WILL brick. There is honestly no possible way to know for sure what is up with the device.
If I had to guess, something is lying to you about what you are running. If your current firmware version is from an I9300, then that is most likely what you have. That firmware WILL NOT work on any Qualcomm, US model S3's. This tells me that its probably an I9300, and someone screwed with something, (or was careless when trying to copy Samsung), in the bootloader. Because it tells you it is a T999 in download mode, it will probably not let you flash the I9300 firmware. But if you try to flash the T999 firmware, there is a very good possibility you will brick it.
It is up to you what you want to try, but understand you are taking a bigger than normal risk since you already know the device isnt normal to begin with.
One way you can try to determine what motherboard it uses, (though this is by no means a for certain way to know), is go to the SIM unlock thread in General and try to follow those instructions. Also try the GPS fix thread in the Q&A sticky roll up. If I am not mistaken, both use dialer codes that are specific to Qualcomm devices. If they work, its a better chance of it being a T999. If they do not work, its more likely to be an I9300.
I decided to go ahead with flashing root66 firmware and selected the latest 4.3 can't express how nervous i was to find out the outcome of it and then finally when the process was over to my surprise i did not face any bootloops or any other problem infact the network seemed to be working well and no problems as far now happy !! Thanks for the help @DocHoliday77 @tokenpoke