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spending money to get a new post I might as well look for carbon
You say this, but you also say this:
without breaking the bank
Which is it to be? Cheap carbon posts are a waste of money, often heavier than aluminium, usually with shit clamp designs and always with the clamp glued to the tube, a potential point of failure.
Pro Vibe 7S is a good mid-price option, a good 100g lighter than my usual suggestion of "stop being a weight weenie and get a Kalloy SP-248" :-)
Why? It's not the post material that's causing the click, it's the rail clamp design.
Are you looking for straight or set back? What diameter? What length?