I suppose this is one of those examples that we see of evolution/survival of the fittest in action. House Wrens are more common because they are more aggressive. I wouldn't be surprised if the gourd house was just too close for comfort for the House Wren to tolerate.
The only suggestions that I can make are 1) just keep disrupting the House Wrens when they start to nest that they abandon the whole area, although it is doubtful that the Bewick's Wrens will come back this year, or 2) hope that the House Wrens don't come back next year. I have a house that House Wrens used for three years (the early brood for two years and both early and late broods for the last year). Then last year, the male showed up and sung his declaration of the territory, but there was no nesting done at this site and he disappeared. He has not been back at all this spring. Who knows why, since the broods all did well and survived to fledge. All I can figure is that his regular mate did not come back last year, and the new mate he found wasn't interested in that house. But that could also happen for you at some point.
Too bad that everyone just couldn't get along! Bewick's would have been nice to have nesting.