Can you take a high-resolution picture of the area around the battery and lightning connector on the board, and while at it make sure nothing is missing there? If there is damage in that area, it could be causing this.
Also typically, when you connect the charger and phone starts bootlooping, it could be that it is not seeing the battery. Could also be due to a missing or damaged component.
If you see nothing missing, another reason could be because the power button is damaged. Disconnect the power button flat cable from the board, plug in screen, plug in battery, and connect to the charger, what happens? Does it still boot loop?
I cannot see from the picture that something is missing. At this point you have to rule out corrupted NAND/software. I hope you don't have any important data on your phone, and that you can replace the Power-Mute-Volume cable.
Use the battery and connector that you think are functional, and use them to put your iPhone in DFU mode, NOT restore mode: connect your iPhone to your computer and start iTunes. Then turn off your iPhone. Wait for a minute so that it is completely off. Now press and hold Power Button 1 second, keep pressing Power Button and now also press and hold Home Button. Wait for 12 seconds, then release the Power Button. Keep pressing Home Button another 15 seconds until iTunes tells you that it has detected a phone in restore mode. At this point choose to download and restore your iPhone.
Alternatively if you don't want to replace the P-M-V cable yet, you can try from restore mode although I am highly doubtful it will be successful if your firmware is corrupted: start iTunes on your laptop. Turn off your iPhone. Press and hold Home Button THEN connect your iPhone to your laptop. Keep holding the Home Button until iPhone starts and shows you the "connect to iTunes" cable and/or iTunes tells you it has detected a device in restore mode. Choose to download and restore from iTunes. Hope this help.
If you rule out bad firmware, and your battery and dock connector are also confirmed good, at this point it becomes a guessing game OR you nicked something off the board somewhere during your repair attempt and it needs to be examined using good magnification; voltages and power consumption have to be measures, etc.