Hi Edward,
No relation that I know of, even though we share a last name. :)
Anyway, to answer your question, I'd go with the lightning connector removal first, then skip to Step 23 of the Battery Replacement guide and follow through to the end. The advantage to doing it that way is you have a clear path for removing the battery adhesive strips and can pull them out flat rather than having them bend up to clear the speaker and other parts that you'll be taking off for the lighting connector assembly replacement.
When putting everything back together, go the other way and put in the lighting assembly before the battery; that will give you more room to work in placing the parts for the lighting assembly. Save putting the battery in for last.
Pro tip for the battery: Just before sticking it down with adhesive, plug it in first. That way you avoid the pitfall of having the battery stuck down in a position where the connector doesn't reach where it needs to go on the motherboard. I speak from experience here, after cursing myself for having to remove the battery again. Be sure and unplug the battery again once it's stuck down before continuing on to reinstall the display.
I didn't put in links to the guides you'll need since it sounds like you've done this before and know where to get them from, but if I've assumed incorrectly let me know and I can add links.
Good luck with your repair; let us know how it goes!
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