TEA is pretty easy to remove using a rotovap if you turn up the bath and use patience. Alternatively, you can distill it off of your product.
The pyridine/copper sulfate thing was always a pain in the butt for me but some of my coworkers swore by it. At least it makes pretty colors.
If I had your workup to do, I'd use ether as the solvent (amines love DCM) and assuming your tertiary amine isn't too polar, wash something like 5x with water, then brine. Then I'd pull off the remaining TEA with vacuum (bp 90°C).
Also, just use 1.1 equivalent of TEA rather than an excess and filter off the TEA-HCl salt before workup.