I ended up using this Add-A-Circuit Fuse tap, which goes in place of your original fuse, instead of the cheap fuse tap, which goes over your fuse. It was more expensive, but it worked really well!
For those who are curious, I added it to the radio accessory circuit that turned on with the car. It effectively split the power so the device on the original circuit continued to work in addition to powering my transmitter (you have to add two fuses to the tap, the one parallel to the wire is for the wire).