Run the logic in the chip at 5V, same as Picaxe. No faffing about needed there for interfacing.
What you choose to supply the display elements with can be different.
That I think is the problem; that there is no obvious separate power supply for chip (logic) and the displays, or, perhaps more accurately, when driving four separate 7-segment displays using MUX1 and MUX2, there is a question as to whether those MUX outputs can readily be used to control the external high-side transistor drives for the displays if that supply is separate and higher than the chip (logic) supply rail.
For controlling just two 7-segment displays in static mode, where the device simply sinks each display, it seems the chip could be run at 5V with the display drive voltage higher, but again that isn't detailed in the datasheets. And isn't what the OP desires.
I suspect it is possible to run chip (logic) and display drive from (upto) 15V and simply connect SDA and SCL to a 5V PICAXE with pull-ups to 5V as Allycat suggests in post #3. The datasheets however, as noted by the OP, are not clear on that.
One arguably needs level-shifting when interfacing a 3V3 system to 5V system using the I2C bus and the same can be argued for interfacing a 5V PICAXE to a 15V driver chip.
Added: However; looking at the datasheet SAborn provides a link for in post #8, the SDA and SCL voltages are defined as absolute 0V to 5V, < 1V5 low, > 3V high, rather than with respect to VCC. That would suggest to me that as Allycat suggests in post #3 would work.
I think it's going to come down to trying it, seeing if it works.