The thing is, Gore-Tex adds $15 to $25 to the cost of a boot, putting them above your target= price. About the closest thing I'm aware of these days is the L.L. Bean Leather Cresta Hiker, which is as you described except for the price: $169. But that's pretty darned close! And at that, they're a bargain. Another boot that's really doing well on the market, and is earning a lot of praise, is the Asolo FSN 95 GTX ($150), which has a Gore-Tex liner. But the construction is of the pieced-together sort, albeit still pretty good. You might take a look at that as well.
Otherwise, my suggestion is to skip the Gore-Tex. Really, good-quality boots, when well maintained, are about as waterproof as possible when you take into account foot sweating, water over-topping, and everything else. REI, for instance, sells a really nice boot called the Spirit V, which as one-piece leather uppers and sells for $155. There's also the Vasque Renegade, which sells for $150 and has one-piece leather uppers and polyurethane midsole for support and comforta nice boot for the money. And really, occasional treatments of a good leather waterproofer will keep these boots as dry as if they had a Gore-Tex liner. Drier, if you consider that most hiking takes place on dry days, when sweating will produce more moisture than the boot could possible absorb. And the simple fact is that a boot with a Gore-Tex liner does not breathe as well as the same boot without. There's just no way around it.