The Hopper3 uses what is called an M10 round connector (I believe). You best bet might be to strip a piece of RG6 wire down to the copper core, cut it to approximate the length of the provided antenna and put it in the center connector of the M10 port. Since the antenna is already at 90 degrees, and adapter probably won't gain you much.
It sounds like you are putting the H3 in a confined space. I'd cut the back out of the cabinet at the point where the back of the H3 sits, otherwise you are going to experience heating problems in a confined space with doors on the front and a closed back. I'd also put a small fan of some type there to suck hot air from the area of the right side of the H3. Another possibility is to remove the door or put the H3 on top of the cabinet. It needs to be in open air so it can dissipate heat efficiently.