Unless you plan to remove the chip for programming you have to dual port the memory chip, easy enough but adds several chips. I had an EPROM emulator on the market that plugged into the EPROM socket and used SRAM with the dual porting added.
I had though to remove the chip for programming but I see there are I2C bus expanders with tristate outputs so I'd probably use those to dual port the memory.
Well these old chips weren't very fast eh? Maybe connect the AVR (or whatever) up with pins the same as an EPROM and monitor the CS/RD signals, when you see an active select you use the address to index into an array and provide the data.
This was proposed by somebody here quite some time ago, originally I poo-pooed the idea but it may have legs. Especially if you can control the target CPU's clock.
The NMOS version of the 6809 couldn't tolerate a stopped clock but you could certainly stretch it. If the AVR supplied the clock, it would be pretty easy to do what you suggest, I imagine. An interesting exercise.
Adding dual-ported RAM would be easier but uses a few chips, this idea only needs a single chip.
Albeit one with lots of I/O. 
Or build an FLASH emulator for development and drop a chip in when finished, just like we used to do but without the UV light
I had two big EPROM erasers running full time. I'll never forget the smell of conductive foam exposed to UV light!