The lightest bullet commonly available is a 200gr bullet made for the 375win. If you choose that for deer, then keep it around the same vel for best results. You cna push them very fast, but they aren't designed to stay together at higher impact vel. The 375h&h is a very versatile hunting round.
"Given adequate penetration, a larger diameter bullet will have an edge in wounding effectiveness. It will damage a blood vessel the smaller projectile barely misses. The larger permanent cavity may lead to faster blood loss. Although such an edge clearly exists, its significance cannot be quantified".