The thing I hate about CT and flap use is that the CT trims for speed, so as you slow down it feeds in more and more "up" elevator, until it eventually maxes out at full "up".
For proper trimming with flaps extended, I generally need a little "down" elevator, so full "up" elevator is a serious detriment.
As far as extra workload for the pilot, to use manual trim I essentially have zero extra workload, so that isn't a factor. I enjoyed the benefits of both CT and manual trim, with no negatives. Manual trim riding the edge of a stall easier and more predictable, and improved my ability to make the shots I take.
CT also trims for wings-level, which is an attitude I seldom enjoy in a fight. You'll find me banked all over, and upside-down a fair amount of time. While upside-down, I find full "up" trim to be more detrimental than helpful...
To eliminate extra workload, but still benefit from manual trim, I simply map a button to toggle the CT on/off. An important part of that though is to set the CT so it's OFF by default, rather than on... That way, if I make a mistake or bump something in a fight, the CT reverts to "off". Suddenly, accidentally, having CT revert to "on" got me killed a bunch until I figured that out...
Next, I use an X52 stick, so I just map my trim to the rotary dials. I then manually trim for straight/level at around 325mph, and use a fine-tip sharpie marker to mark my rotary dials (I only fly one plane of course, so trim settings never change for me...).
Once my trim dials are set, I manually turn CT ON for most "general" flying. Anytime things get to a non-general flying situation, I just bump the CT off. My manual trim automatically moves to its pre-set setting, based on my dials. It does it the same every time, and it doesn't matter what my speed is. My plane always "feels" the same, and I get a much more precise "feel" for my speed, maneuverability, etc, because I'm not having those things distorted by constantly altering trim (which, as mentioned, is often trimming detrimental to my flying habits anyway).
So, essentially, my extra workload in a fight was tapping a button with my thumb.
In the event my target needed a bit of chasing and the speeds got back up (and mostly wings-level), I just tapped the button again to turn CT back "on" until I caught him and forced him to turn.
Even with flaps up, those pre-set trim settings make roping and either shooting or reversing at the top off the rope "oh so nice" too.
As a side-note, those manual trim settings also make landing a breeze. Tap the manual trim button to allow the pre-sets to take over, drop flaps and gear, and the plane flies almost hands-off during approach. And if you need to trim for a missing chunk of wing, that's no issue either...