A PICAXE solution is potentially possible, but I suspect it would be much easier to timestamp the data in the PC.
Presumably you will have some PC software which would read the RFID data and the timestamp received and store it away on disk or otherwise process it, so it shouldn't be that much more difficult to have the PC software add the timestamp itself.
This is an easier and potentially cheaper solution than adding PICAXE's - A PC executes about 100,000 times faster than a PICAXE, PC serial has buffering so much less chance of missing data, and PC clocks can be set as accurately as using a DS1307 with NIST or NTP servers if you have an internet connection.
If you don't have an internet connection and the PC's clock isn't reliable enough, it would be possible to have a PICAXE handling a DS1307 and create a simple hardware mechanism to allow the PC to select between the RFID data stream and the PICAXE on a single serial line, or even more simply, put that on a separate serial port. The PC clock is probably accurate in the short term, so the DS1307 only needs reading when the PC or RFID handling application starts up.
There are all sorts of problems in using a PICAXE to handle back-to-back and high-speed serial data, and especially so when asynchronously receiving and transmitting, and you would probably be creating a lot of unnecessary, difficult and time-consuming work using PICAXE's to do this. There's no guarantee that what you want would even be implementable using PICAXE's.