Beech is fairly shallow rooted, so removing the bulk of the root ball should be fine. You need at least 4"/100mm of concrete, if your van is frequently heavily loaded I might be tempted to go to 6"/150mm, possibly with reinforcing mesh at half depth. Fill to the bottom of the root ball holes with concrete or backfill with rammed MOT Type 1. Don't backfill with soil or a void will develop and the concrete could crack.
I would just prune back the lower branches regularly.