Should I Do an at Home TCA Peel for Hyperpigmentation on Legs?

Despite the fact that you have obviously researched the topic, understand the basics and are on the right track conceptually does not mean you should play doctor. "A person who serves as his own lawyer has a fool for his attorney". The same can be said for medical treatment. You are at high incidence to not have a uniform result and working with a Dermatologist or Plastic Surgeon is essential as  prescription medications, intervention and titrating the TCA strenght are often needed rather than a simple "cook book approach" you are suggesting.

That being said here are some general comments: It is important to pre-treat with some prescription creams first and use the same products when your plastic surgeon or dermatologist allows you to use it again. This is to prevent brown spots following a TCA or any other peel, laser or deep dermabrasion procedure. This is  called post inflammatory hyperpigmentation. The inflammation part is a normal part of the healing process but unfortunately you are sensitive to the sun which is what causes it. Ideally your doctor would have pre-treated your skin some weeks before your treatment with Retin A, 4% hydroquinone, and a strong SPF sunscreen and/or other agents to prevent this from happening. As soon as you are given permission you would again start these products plus perhaps Kogic acid and use them for some time

In general, I do not recommend home peels. Brown skin, Asian skin, East Indian and African American skin is prone to hyperpigmentation.
Before you spend $$ on peels or IPL, try these topicals.
This is called post-inflammatory hyperpigmentation (PIH). 
I recommend Melarase AM and Melarase PM creams from Kare Skin as well as a broad spectrum sunblock like Spectrase. Bothe are from Kare Skin. Scler-x post inflammatory hyperigmentation relief complex is a great supplement to take as well. See link below. 
Spectrase is a wide spectrum sunblock SPF 50+that is also recommended to prevent ongoing inflammation.