Are you saying that you already have the georeferencing file? I'm familiar with world files, which have extensions like .tfw, .pnw, and .jpw, and are used to relate an image to geographic coordinates. I've never heard of a map file for this purpose.
I found something called Viking which might serve your purposes. I've never used it, but it sounds like it might check your boxes.
You could also use QGIS with the Georeferencer GDAL and GPS Tools plugins. If you can output your maps in GeoTIFF format, or if you can output a world file, you can open the maps with the software and get right to making the GPX. If you can't get the georeferencing imported, use the georeferencer plugin.
The pictures don't indicate that it's broken AFAICT, but obviously it could be in subtle ways. The off-center degree etching on the plate is centered around the retaining screw for the clinometer, which is the black assembly. I've seen this referred to as a geographer's compass; here's a more modern example.
You didn't ask, but I wouldn't use that for orienteering; while a beautiful piece of history, it looks very heavy and appears lacks a good straightedge or a ruler.
You could do that with a map, it just depends on the type of map you'd use.
​
For an app, my personal go to would be Mapy.cz with its Tracker feature. Don't worry, the app can be set to english, but the images are in czech.
You can take the full size image and convert it to a pdf. Then use any number of online sites for collation.
I've used https://rasterbator.net/ for printing out poster projects when the UW wanted to charge me 50 bucks for it. (The link looks sketchy but I promise it's good). That site works well and is easy to do. Play around with it once or twice on your home printer just to understand a few settings. But it's great for scaling.