Hmm, I'll do a bit of math with my own monitor. As I do not know the physical dimensions of yours.
My monitor displays at 1280*1024 (it's almost square). For the sake of simplicity, I'll just take the width of the screen, since the height will (AND SHOULD!!) yield the same number anyway. It's 1280 pixels WIDE, and if I grab my tape-measure (which luckily also has inches on it, so I don't have to convert cm to inches), my screen is 'about' 15 inches wide. So 1280 pixels, divided by 15 inches, is 85,333... So the DPI of my screen is just over 85.
Let's do the math, your file is 7200 pixels by 7200 pixels. So on MY monitor your full res map would be just under 85 inches by 85 inches... (so I'd see only a small portion at a time of your map. Now, the more pixels per inch, the smaller your image gets. Because those 7200 pixels are a CONSTANT. If you raise the dpi, the physical size of the image in inches will be smaller. If you lower the dpi, the size will increase. But also remember that dpi also measures the amount of detail per inch.
I don't know how the 'quality' of your print ended up wrong, but this can have several causes:
1: Your colours are 'out of gamut'. Your monitor can show colours your printer cannot show. This only applies if the colours looked wrong.
2: Your printer is not fit for this job. This is unlikely, since I have a cheap multifunction printer myself, and it prints fine.
3: You have detail in your map which starts to look bad if it is compressed into a high DPI, this is quite likely, since you say that the large test-prints at the professional print-shop turned out 'reasonable'. 7200 pixels printed at 150 dpi is in fact the same result as printing your 300 dpi image 'on 200%'.
So, look up the current resolution of your monitor.
Grab a tape-measure, and physically MEASURE your monitor.
Divide the number of pixels by the number of inches. That is the dpi of your monitor. If you want your print at the exact same size. Print at that dpi, but be warned, paper doesn't tolerate 'low' dpi's as well as monitors do. Your monitors dpi is probably around 3 times as low as the dpi you're asking from your printer. Which also means the image on screen appears 3 times bigger than on paper.
I find that for gaming purposes, usually in rooms which have been lit 'satisfactory' and people just admiring the map from about 30-60 cm away (1-2 feet), 150 dpi is fine. 300 dpi is REALLY high quality, but I doubt people would see the difference without an actual side-by-side comparison.
Try to keep print and screen separate in your mind. If it helps, you can also activate the on-screen rulers in GIMP or Photoshop. This way you can get a good feel for the size of the PRINT.