Looking for 4 bit memory option
It seems that you want to build as much as possible yourself.
So why not also build your own the ROM?

It is easily possible if you have just 2^4 data words.
You can use two 74HC138 to decode the address and feed the outputs into a diode array.

Here is a schematic with just one 138 (but another one can be added for 8 more data words; it must be enabled by connecting A3 to the non-inverted enable input; another possibility would be to use a single 74HC4067 (16 channel analog MUX/DEMUX)). 
A diode has to be placed at the appropriate crossing for each 0-bit.
(If you have more 0-bits than 1-bits use a 74HC240 buffer instead of a 74HC244 and put a diode for each 1-bit).