Don't think they would tunnel but would bridge taxiway Alpha just to the north of the road tunnel with the station being on the site of Easyjet's HQ. If Gatwick is anything to go by it is not really a station so the footprint required is quite narrow.
I also think the other end would be on the Kimpton Road side so they would not have to cross the railway.
The shuttle bus link is truly shocking at the moment with only 1 bendy bus in service. Two others are broken and the fourth one is having new seats fitted. The missing three bendy buses have been replaced with low capacity single decker buses that are often crammed full.