"We have but a choice to be responsible citizens"
"Have but ___" is a way of saying you have a said amount/number/degree of something. It is a phrase that in standard use implies scarcity.
- I have but one chance left = I have exactly one chance left
- I have but seconds left to live = I have seconds left to live
Although the second one doesn't give you a definite number of seconds, the phrase implies that how many ever are left are clearly not adequate.
So your sentence does mean what you intended it to, but your instinct is right in that it sounds just a bit clunky and obtuse.