if you just fly a heading to compensate for the drift, you will have to have a hefty angle of bank on the inbound turn to avoid flying through the inbound track
No you won't.
Simply continue the turn, rolling out on a heading to intercept the inbound track. Just like the overshoot that occurs on a Sector One (parallel) entry.
Remember you don't have to be "on track" during the inbound to the aid anyway, only after commencing the let down and even then the AIP was changed a few years ago to allow a descent to commence so long as the aircraft is on an
intercept heading for the inbound track.
No mention is made of how far off track you may be at this time!