depends on a number of factors such as distances involved and how committed the BE school is to the kid. The players he has listed may be the result of as few as 3 or 4 scouting trips. It doesn't take much to get listed at higher prestige schools, so don't assume he's spent all his money just because he shows more considering than he has spots.
I personally don't think the prestige ratio is as high as you have it, but you will need to put in considerably more $ to equal the same effort. Not sure how the BE did in Smith, but generally the big 6 get at least the equivalent of an extra scholarship of post season cash, often much more. With his prestige advantage and potentially at least similar if not more cash, even a distance advantage wouldn't generally be enough, especially since it look like you'd be picking the battle. If FSS shows him tight with at least 3 other kids and not 100% sold on the kid you like maybe that's an indicator the kid is a backup option, but that's far from a surefire assumption.
Realistically I'd say you have little chance. I'm not saying it can't be done, but (and I mean this with no rude intent) if you have to ask about the possibility its unlikely you are battle savvy enough to pull it off.
A last bit of advice - if you do go forward with this (or even just to protect what you already have) its generally a not great idea to tell everyone that reads the forum how much you've spent and how much you have left. Knowledge is power in recruiting and the less sure info you provide the other guy the better off you will be. GL.