I buy a lot of iPhone aftermarket batteries. I estimate that 25% are “poor” when new. 75% are “very good to excellent”. I was buying great batteries from a specific vendor, and suddenly they changed suppliers I ended up with a bunch of duds that couldn’t hold a decent charge. So beware that someone else’s experience with a vendor may not be the same as your experience. Similarly, not all battery adhesive strips are the same. Some are good, but some of these adhesives are made so poorly that you’ll never be able to release them properly. I got a batch of bad adhesive, and battery re-replacement became a huge time sink as I spent far to much time trying to release the battery, followed by more time trying to remove the sticky remnants of the lousy aftermarket adhesive.
It’s definitely an easy job to replace an A1342’s 2.26 logic board with an A1342’s 2.4 logic board. No other component needs to be swapped or modified. I’ve done this upgrade to many “late-2009” A1342’s with damaged logic boards.
Error 53 is a vestige of history. It is no longer a problem since Apple released iOS 9.2.1. As long as you're not running iOS version 9.2.0, you shouldn't see error 53. See https://techcrunch.com/2016/02/18/apple-apologizes-and-updates-ios-to-restore-iphones-disabled-by-error-53/
My first guess is that a cable on your new display is unreliable: As the cable flexes when you close up your iPhone, one or more conductors in the cable are losing their connection. (1) If evidence shows that there is an iffy conductor, I'd remove and reattach the display, in hope that it's simply a poorly seated connector. (2) To determine if it the simple case that the display is simply not displaying anything, I'd try using a USB cable and see if the device shows up in iTunes even with a non-functional display. (3) If the old working-but-cracked display is still available and viable, I would temporarily re-install it and see if the old screen's behavior is normal.