I think if the list is really down to the 4 names which it appears to be, at least to me, then White has to be the guy. The 4 names being: Archie Miller, Chris Mack, Donnie Tyndall and Michael White.
Chris Mack is making around $1m/yr and his buyout isn't known, but it's thought to be around that same number. He would likeky require at least $1.5m+ to leave his alma mater where he has things pretty good currently. Add in a decent salary pool so he can surround himself with good recruiters, something Hart has made clear he wants, and you're talking a pretty decent investment.
Donnie Tyndall is only making around $500k/yr, with his buyout being close to that same amount. In theory he would appear to be cheap, but I'm not so sure that's gonna be the case. Tyndall is no dummy, and he realizes he's likely 2 years away from landing a better gig than Tennessee(assuming he keeps winning). Just speculation on my part, but I think he would likely want around $1.5m/yr, at least a 5 year deal, and a steep buyout to cover him, but low in terms of his ability to leave.
Archie Miller is hard to gauge because nobody really has a clue what his contract numbers are. He got a new deal in February which was before a tourney run made him such a hot name, so it's not hard to imagine that his deal isn't anything crazy. I would say it's possible though their AD front loaded a buyout to prevent it being easy for him to leave after this year or next. I would venture to guess it would take similar numbers to Tyndall, around $1.5m/yr at least to really get his attention. I'm sure he also realizes that he's another good season away from getting his phone blown up with better offers than Tennessee.
Michael White seems like the most attainable to me. He's a young guy that seems to be ready to prove himself at a bigger school than LT. He's now been there 4 years so he has that on Miller or Pitino. He's making $600k with a buyout around $400k, so very attainable from a monetary standpoint. I think he's likely the coach you could get for the cheapest, as LT likely wouldn't put up the fight Dayton, Minnesota or even Southern Miss would. I would think you could likely land him at around $1m/yr, which would be an absolute bargain if he's able to produce results.