If I've found the choke that works best in my 870 and I have it back bored, will the same choke still be the best or do I have to start patterning all over agan?

Anytime you make changes you'll need to start over. It make still like what it did before. But there's no guarantees.

Backboring increases the diameter of your bore. I doubt if the same tube would fit after it's backbored. Try just having the forcing cone lengthened first. That is often enough to improve your patterns and won't affect the choke.

Try polishing the forcing cone and choke first, you might be surprised at the difference it makes.

YES!!!!!!!!!!!!!!!! this one makes me a Rack Buck!

congratulations timberjack82 !!!