Can you post a photo so we can see where the pressure mark is? If the battery is too far to the left, or too high up, then the screen will get caught as you go to close it, causing pressure marks. If it looks like the image below then it should close properly, unless the battery is defective, or there is a bend in the phone chassis.
Certainly looks like a faulty/damaged LCD. Try a replacement screen to see if that solves your issue. The repair is easy enough to attempt yourself, just follow this guide: Substituição do conjunto do painel dianteiro do iPhone 6 The iFixit screen is around £60, so it is also going to be cheaper than replacing the device. All tools and parts you’ll need are listed on the repair guide too.
Buy a cheap, broken 6s with a relatively good condition housing, then simply swap the board over. The repair on a 6s is relatively easy to do, especially if you have previously repaired your device. If you aren’t sure what to do/what order to remove parts follow this guide: iPhone 6s Rear Case Replacement (1) Bonus of purchasing a broken 6s, is that it may have working parts inside you can use, which would speed up the repair. I don’t know why they think it would take 3 hours, it can be done much quicker than that,
The cable comes soldered to a replacement charging port. Order another one of these and you will be fine. https://eustore.ifixit.com/en/Parts/iPho...-Parts/iPhone-6/iPhone-6-Lightning-Connector-and-Headphone-Jack-Fix-Kit-Black.html?utm_source=iFixit&utm_campaig...
Try a hard reset first (See: Comment forcer le redémarrage de l'iPhone 8) However I think you are correct, a software fault is the most likely cause of this and that would be fixed via an update/restore. Is the customer on the beta program? A friend of mine is and he could only answer incoming calls, his dialer wouldn’t work until the following update was available.
Was there any power running through the device when you disconnected the original battery? It’s always worth double checking the connector to make sure it is sitting down correctly, and ofc use an Mfi cable. If that all checks out then you could have received a faulty battery, if this is the case there should be a warranty and you should be able to get it replaced.
It is worth taking a look at it with a torch or something similar. Try and look for anything which could be blocking your charger from connecting. This can be something simple like a ball of fluff, or something a bit more gross. If you do see something in there you have a couple of options.: 1) Take it to a local phone repair shop who will be able to clean it out with tweezers, and also compressed air. (The good thing about this, is that a reputable shop would be responsible for any damage caused) 2) Give it a go yourself, be aware that they are very fragile and if you use something too thick it will break. You will then need a replacement port which is considerably more expensive. https://www.youtube.com/watch?v=R6p0Eg-yq3A <— Video on someone cleaning out their Micro USB Charging Port.