Fawkes' carry weight is... odd. According to the companion wheel, it's only 165 Lbs.

The Fallout 3 Wiki (And I know this isn't Fallout 3 technically, so this wiki is hit or miss for accuracy, but) says Fawkes can carry 229 Lbs of gear. Unless there was a nerf to him in TTW that I forgot to read, he's a bit weaker than before.
He's carrying roughly that for me right now. However, I can't get him to carry it by just giving him things normally. If he's "over-encumbered," I need to take something heavy from him, before giving him more things, and then returning the heavy item to him. He is now carrying more than he was before we traded, despite being "unable to carry anymore."
He couldn't carry anymore. I took back my missile launcher. I have him 25 Lbs of various missiles. I gave him back the missile launcher. And the wheel still says it's only at 165/165 Lbs.
