Lucy68 said: It does look dire but the branch ends look healthy. I wouldn't do any pruning. If the branches actually do die, you can prune them back but let the tree leaf out next spring to determine how much pruning you need to do.
Your winter plans sounds good to me. Unless the lime is a key lime, both your trees will be fine to about 25 degrees.
Lucy68 said: Key limes are not as hardy.
Some suggestions for winter... keep the temperature between 50 and 55, over 55 would be ideal.
Citrus need humidity but keep the air moving or you will be fighting mold and mildew.
Don't forget to water.
Hopefully you have good lighting. I would be using full spectrum LED lights rated at 2500 to 3000 lumens left on for 8 - 10 hours a day.