Here is the formula for RPI:
RPI = (WP * 0.25) + (OWP * 0.50) + (OOWP * 0.25)
where WP is Winning Percentage, OWP is Opponents' Winning Percentage and OOWP is Opponents' Opponents' Winning Percentage.
Assuming we win two tourney games, the first part of that gives us a 0.0064 improvement.
Best case scenario for RPI is that we play Ole Miss and then Vandy. If I assume that the rest of the rating is evenly distributed between OWP and OOWP, I can estimate the second part of the formula above as improving about 0.0026.
For the third part of the formula, Vandy has the 6th toughest schedule and Ole Miss ranks 36th. I don't know how to put a number on that though, but using the 6th and 36th rated teams win % in RPI gives a rough idea, and the improvement ends up being 0.0036.
The total improvment is then 0.0126, which makes our RPI rank about 58-59, which might be just good enough to slip in. (Let me add that I am assuming that the changes in teams below us and above us cancel out, which might not be true).
Now the thing I leave out is our last game against UK. Just PLAYING UK will be a huge improvement, because UK has the best win percentage in the country and that's half the formula. In other words, even if we beat Ole Miss, Vandy and then lose to UK, our RPI from the UK game will go up a bit, which hopefully cancels any inaccuracies in my calculations.
Of course, if we BEAT UK, it's all a moot point.