With that kind of budget, you can build a pretty superb rifle.
I'd start by looking at a Model 600 Remington. Very small action. Very well-suited to .243 or even .223, which can be a great cartridge out to several hundred yards.
You have the budget for barreling, stock work, etc. A nice small action is a good start.
A few years ago, "Scout" rifles were pretty hot. Ruger made a good one, but over-priced for what it was. Good potential, too. And Remington Model 7 was a nice little rifle, right out of the box.
That said.. .and before we settle on a bolt gun... what about an AR platform? There are a ton of options and, built right, the AR can be a great choice for a small-statured person. Lots of options there.
Hey
Tucker301 How about moving this into Bolt Action or Gunsmithing?
Look up the 'scout rifle' threads from a few years ago. I can post a picture of my Model 600 Scout custom if you like. Fun project.
Cheers and sounds like a great project,
Sirhr