The problem is that when you connect a cable into the channel insert, it disconnects the rest of the signal path *unless* you provide a return path. That's why you are getting either one or the other.
Some boards allow you to put a TS cable into the insert jack only partway, and tap off the channel signal without "breaking the normal" and disconnecting the signal. I don't know if the Mackie is one of these, or not. It is worth a try, and might give you a short term solution: this is called a "first-click" insert steal. The plug only goes in until the first click, about 1/4" from being fully seated.
That is kind of unclean, though. The right way to do it is make some specialized "insert-stealing" cables. These use a 1/4" TRS plug on the mixer end, and a regular 1/4" TS plug on the other. These are not a normal balanced-to-unbalanced adapter cable: their only use in the world is stealing inserts. Nobody sells them premade, so you'll need to make them. And paint the connector shells red, or something, so you know that they are to be used *only* for this, when you're digging for cables...
Here's the secret. In the TRS plug, short tip to ring, and connect the hot lead of your cable to that shorted node. The shield goes to sleeve, as always. On the other end, it's just a normal TS connection. By using a TRS jack with tip shorted to ring, you can insert the plug all the way, and the short connects your new direct out to the insert's return input for you. This lets your external electronics bridge across the normal signal path, *instead* of interrupting it. So you can do anything that you'd usually do with the mixer: do effects submixes, monitor, yadda yadda. It's a moderately clean solution to a dirty problem.
On an insert on most boards, the channel signal send is on the ring, and the return is on the tip (but not all- check your board's doc). Once a plug is in the jack, this normal is broken. The shorting link restores that normal connection. You can't use an off-the-shelf TRS balanced-unbalanced cable for this, because they usually short the ring to ground- which is most definitely *not* what you want. And you do not want to mix this up with a bal-unbal adapter cable, since it would be a dead short from + to - if used in a balanced application... Thus, the red paint on the shells (when I make 'em, anyway!)
Caveats about using a insert-stealing approach: there is no buffer amp between the external tap and the internal signal path. Use as short a cable as feasible, and be careful with your grounds. If you load down the tapped signal you will affect your levels for that channel within the board, and can possibly run into distortion, frequency response, or noise-injection problems and so on. But still: this is by far the easiest way to mimic a real direct out that there is.
Hope that helps.