It is hard for me to envision the problem you are having without seeing what you mean. The image above was made exactly with the setup above. 2 750ws flash heads dialed down to 1/4 power, f18 and a shutter speeed of 1/250. Pretty straightforward. After running the test, I left everything as it was, posted the shot and then went about my invoicing and shipping for the day. Later, when I was asked for a shot of the setup, I just grabbed another camera and shot the light tent without any changes at all. In fact, the same knife was still in the light tent.
The door I'm talking about will just shade the front of the light tent from extraneous light leaking in the front. My old one encircled the product 360 degrees and had some holes in it for lenses. This one is open at the front which is what motivated my desire for a light shield for the front.
I doubt flash or continuous lighting is an issue. I use flash because it is fast, predictable, daylight white balanced, more powerful, cheaper to operate and free of heat buildup. Photographically, however, the shot should look the same with continuous light. The exposure and the white balance would be different but that shouldn't be an issue.
Don't get me wrong, photographing knife blades or anything light or shiny against a white background can be tricky. You need to fool around with light placement, subject placement, exposure etc.