I doubt it would affect your bonding very much, but the stress from travel could potentially put a bit of strain on him. A rabbit neuter is a pretty simple affair though, if the rabbit is healthy and reasonably young. (With an older rabbit you might have to warm the bun up post surgery with a heating pad, sock with rice, hand warmers or a warm water bottle, and also syringe feed Critical Care or soaked pellets the first night back home. One year is still young though, but it’s good to be prepared).
To help his tummy during the travels, you could give him a probiotic. The most bunny appropriate ones that have been scientifically shown to make a difference contain saccahomyces cereviciae (brewer’s yeast) and/or enterococcus faecium.
Most buns won’t eat hay during travel, and water could also be a problem. Fresh herbs can be served in the carrier as an appealing alternative.
How quick a bun warms up to its human varies a lot with the bun’s personality. Rabbits are wary by nature, but more or less so. To make him get used to your voice you can read aloud to him. I read Brideshead Revisited to my current bun Vilde when he was new with me. (That made him demand I supplement his wardrobe with a formal dinner jacket and he also craves champagne with his meals, so maybe I should’ve chosen something less extravagant to read 🤣)