You could probably figured out which direction the unwanted light was coming from and block it with an open hard back book (that would make portable) or tape some white paper in a half circle to diffuse the light as it come in. You might put some dark paper or material on the base to keep light from reflecting upwards. Do you take you images off the coin tube roll? If so, you could bet an old pharmacy bottle like and glue that to the top of the tube to make a bit more room. Put dark material on the lid and you could shoot from there. You could also make a ramp with a 2X2 to direct might reflection of light into the microscope tube.

Light from the source onto the coin and up into the scope barrel. On my setup the wider end of the ramp need to be in the correct position as it will make the images dark. But when you get it set at the right angle (turning it a bit) you will find the sweet spot and the coin will glow. That is what I can see as something I would do to make you system better. Does the scope adjust on the computer? If not how do you adjust it?