How to re-introduce injured hen?

My black australorp hen was found laying on the floor in my coop almost two weeks ago. She was bleeding from her head and had a big bruise on her neck. As far as I could tell, my other chickens likely beat her up.
I nursed her back to health (thanks to the great advice from members on this forum), and she is now doing terrific and back to her old self again.

My hen does have an existing limp in her right leg. She has had this limp for over a year, which results in her being at the bottom of the pecking order.

Currently my hen is living in my basement in a large crate. The other day I brought her outside to free range with the flock and see how they reacted. Most of my hens towered over her and pecked her neck. I pushed them out of the way and rescued my hen a few times. When I returned her to the basement she squawked like crazy. It went on for a while, but she eventually stopped.

My issue is, my basement is kept at about 80*F, and the outside temperature is about 20-30*F. I like taking her for short trips outside, and she seems to enjoy it too. Except, lately she has been shivering while outside, and I am afraid that the shock of the drastic temperature changes could get her sick.
At the same time, she is a chicken and I hate to see her living in a basement for another few months.

What is the best thing I could do for her? Won't these short trips outside get her sick?
Should she remain in my basement? Or should I set up a seperate area in my coop for her with a heat lamp?

Thanks in advance for any advice.