It could’ve looked bad if Handy had done anything himself, for both the king and for Handy himself. Had he approached Twilight in private, it would have looked like he was crawling back to them. Had he interrupted them on stage, it would have appeared all kinds of bad, so he did the best way he could: he left it in Joachim’s claws to spin it to his best interests. To the Equestrians, it looked like he put the pressure on Handy, regardless of whether they had made the request of the king himself or not, and it kept up the fiction Handy was under the griffon’s strict control on a public stage.