I would leave the frozen/thawed mouse in his tank overnight. Just put it in front of the hide he's in, and leave it. As it's dead, it won't harm him, and he may just want to eat without an audience. Ideally you don't want to handle him on feeding day, and then 24-48 hours after feeding, so that he isn't stressed before feeding, and then has time to digest his meal. Honestly, we occasionally handle our boys the day of feeding now (for short periods of time). We've had them long enough, and they've been eating regularly, so we don't worry. But with a new baby, I would definitely leave him alone as much as possible until he's eaten for you a few times.
Feeding in a separate tank will cause additional stress from moving him there and then back again. Most recommend feeding in their tank for this reason. They won't associate your hand with feeding unless you only handle them for feeding (you are more likely to get bitten when you handle them right after they've eaten to take them out of the feeding bin and place them back in their enclosure). As long as you don't smell like food, you should be okay. I feed frozen/thawed on tongs, so they don't associate my hand with food. I chose to do frozen/thawed because it's easiest and most convenient for me. You can safely feed live, but if you want him on frozen/thawed, don't give up after one failed attempt (especially because he was handled today). If you leave the frozen/thawed mouse in overnight, and he doesn't eat it, wait 5-7 days and try again. Offering food too often will cause stress.