“Congressmember” in the generic and “Representative” and “Senator” for the specific houses are widely accepted (standard and formally preferred for the house-specific ones) and gender neutral.
You might be confusing your audience more than you are accomplishing your other objective.
But I definitely appreciate your comment here and it's something I'll continue to evaluate. On one hand it's shouting into the void. On the other... well... it's probably still shouting into the void.