The problem is 18 year old software with no sourcecode that bit-bangs the printer port to drive a 26 year old hardware interface design.
However, if you don't need to access the hardware interface it works OK under Windows NTVDM, and can be used to load, save and edit its .BIN files, which is sufficient to figure out the file format. A very quick look indicates the file may be a simple sequence of 16 bit values, stored in low byte first order, of which the upper three nibbles of each are zero and the low nibble is the contents of that memory module location.
The easy way forward would be to use an Arduino for the memory module interface, and create and publish a sketch that can read and write ICOM memory modules, and transfer data to/from a PC, preferably in a human readable format e.g. ASCII hex dump or IntelHEX, so it can be driven from any terminal program and isn't tied to an OS or even to a particular interface between the PC and the Arduino or similar board. An Arduino UNO should easily be capable of powering the memory module and controlling all its pins with no added hardware other than jumper wires. The sketch should document the interconnections, and also use pin names rather than absolute numbers so it can easily be ported to future generations of the Arduino platform. I believe an Arduino-based interface, with open source firmware will be highly future-proof, as Arduino-like boards are likely to be the de-facto standard for many years to come.
Ideally someone with a fully working rig, + a spare memory module that has already lost its contents would volunteer to do the development, as that would be minimum risk as they could test with the blank module and the BIN images to be found on the web, and confirm normal operation before risking their good module. Otherwise, with a good reverse engineered schematic, + a N.O.S. RAM chip, it should be possible to build a functional clone of the memory board with a little effort.
You'll also need to write a utility to convert the existing .BIN module dump files to your chosen human readable file format.
The uPD444C 1024 x4 bit SRAM datasheet can be found at
http://www.pisi.com.pl/piotr433/manuals/upd444.pdfThe rest of the info needed can be found in this thread, though it would be nice to actually ID that transistor, and get the value of the vertical resistor next to it, or to have a schematic of the RAM board to work from.