WELCOME!
I have a '15 Tahoe LTZ and wanted more off-road capability via increased clearance and tire upgrade, maintain a fairly comfortable ride and still have my 5 yo daughter be able to get in & out of the rig safely
So...w/ the LTZ I have the magneride suspension and that opened a whole new can of worms. You're in a great position with your trim levels for getting your Tahoe up off the ground.
BUDGET - that's a major point here. Have you established a level that you are willing to spend?
After all I have gone through I have decided that my next vehicle there will be no messing around and go with the pros. That is, an engineered solution to my needs. That being said, I have the following recommendation for you. This is a full solution that is engineered for performance.
BDS Suspension
4" Lift -
https://bds-suspension.com/product?ma=5&mo=9&ty=1&yr=2015-2019
4" Lift w/ Coilovers -
https://bds-suspension.com/product?ma=5&mo=9&ty=1&yr=2015-2019
6" Lift -
https://bds-suspension.com/product?ma=5&mo=9&ty=1&yr=2015-2019
6" Lift w/ Coilovers -
https://bds-suspension.com/product?ma=5&mo=9&ty=1&yr=2015-2019
The coilovers BDS is using are FOX 2.5 w; remote reservoir. You might be able to speak with the company and have them use the FOX 2.0 Adjustable Coilovers...that could save some $. IMO, the 2.5's are a bit much unless you are actually going off-road more than 25% of your driving. Or perhaps drive on A LOT of dirt roads in your daily work.
I believe you had mentioned wanting to keep your OEM wheels. Going with the 12.5" tire will probably require backspacing to prevent any rubbing. Either wheel spacers or new wheels. I would say choosing new wheels would make life MUCH easier when doing either 4" or 6" lift. The amount of backspacing needed would really push the outer edge of the wheel past the fender with a wheel spacer, you will certainly be spraying junk all over the side of your truck. And I do believe that wheel spacers can be potentially hazardous, depending on what kind of driving conditions/obstacles you put your truck through.
I really like the Black Rhino wheels - great off-road rugged look. Purchasing NEW wheels would also give you the option of moving to an 18" instead of the 20". That's my final mod on my rig. I think the extra rubber you get from moving down in wheel size has multiple benefits. Better off-road performance via increased sidewall, better handling/smoother ride at speed and over bumps. Plus 34-35" tires on 18s look GREAT on our Tahoes!
Good luck on your search...welcome again and HAPPY TRAILS!