I'm a fan of 3" barrels for a short barrel revolver. They give a little more velocity over a 2" or 2.5" barrel. Maybe not a ton when measured out of a chrono, but enough to make standard load JHPs more likely to expand (though with short barrel loads it may not be as big a deal these days as it was a decade or more ago). They will give you a little more sight radius and be easier to aim accurately. Unlike a small-frame revolver, they are big and heavy enough to be quite comfortable when shooting magnum rounds allowing for quicker and easier follow up shots v. a smaller revolver. They handle and balance about perfectly. Compared to a 4" revolver they shave off a few ounces and they are shorter so they are easier to conceal and carry. Overall, IMO, a 3" medium frame revolver is the perfect all-around revolver and the perfect carry revolver.
My first choice isn't a budget brand, but would be worth it, the S&W 686 Plus. It is in the L-frame so it is strong enough to last a lifetime of any factory loaded or SAAMI spec reload .357mag. In the "Plus" variation, it carries 7-rounds. It is still made. With its strength/durability, extra round v. the K-frame, current production, yet only a tad bit bigger than a K-frame so it is still a reasonable carry size and weight in the L-frame, it blends all the characteristics a revolver like this needs.
My next choice would be a used K-frame. They are a little smaller and lighter than the L-frame so they are even better at any concealment or carry role, while still heavy enough to be comfortable when shooting most magnums. They are a less durable and are better seen as .38spls that can shoot occasional magnums than the other way around, as they will see negative effects of regular .357mag use faster than most revolvers (and if you shoot too many 125gr magnums you can end up with a cracked forcing cone). The biggest disadvantage is that they are getting expensive.
My next choice is a used Ruger Security or Service Six. They are similar to the K-frames but with Ruger durability. They are also less expensive (though there is enough collector interest that I'm not sure they'd qualify as a budget option).
My last recommendation would the Ruger GP100. They are a little on the big and heavy side compared to the K-frame or Service/Security Six, but they are built like a tank and your grandchildren will probably be passing them on to their children and grandchildren. Unlike the S&W, it is only available as a traditional six-shooter, no seventh round. They aren't cheap, but they will be $50-100 less than a S&W.
I know you said you wanted a cheaper brand, and I used to recommend Taurus and Rossi, but I don't anymore. I had a Rossi (made by Taurus) that would sometimes lock its cylinder after shooting magnums, and after some internet research found it is a fairly common problem with current production Rossi and Taurus revolvers. If you want a Taurus or Rossi, I'd look towards one from the late 1980s or early 1990s up to maybe 2006 or 2007 for their best examples.
If you want something cheaper than my suggestions and go used Taurus, they didn't make many 3" guns. You might find a 3" M65 or M66 if you are lucky. I'd look towards the 2" 617 (7 rounds) or the Tracker. I know that the Tracker is 4" and you said you want shorter, but it carries 7-rounds, is reasonably priced, and is on Taurus' "compact" frame so it is a little smaller and lighter (thus more carry friendly) than the typical medium frame 4" revolver.
