Edit the linework in the tap family so it look like the tee - if that is how you want it to present. You should just have to delete a couple of lines and constrain a symbolic line to the main ref plane.
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If it was the tap that created the problem, your solution would be great. Unfortunately it is the system family duct itself that is creating this. I've already edit the tap with the visibility settings. If I pick the tap it only shows as a single line in coarse mode.
If you would be kind to look at my attached picture I hope you can see that in the right picture I hover over the duct and all of it lightens up. It also shows on the tooltip.