You can feed him in his enclosure. There's no need to move him to feed him. You can feed at any hour. My snakes generally eat mid-afternoon because that's when it's convenient to feed them.
There's no harm in trying to switch him to F/T. This site has a ton of tips on how to feed FT. One of the best tips I see is to make sure he is secure in his home first. Since your snake is eating live, you can make certain that he is eating and happy. Then warming the FT prey up properly and hitting it with a little hot air from a hair dryer to make it a nice enticing hot spot helps to encourage the snake to see it as the same as live.
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