The 15 is a 10 with adjustable sight. Fixed don't ever break. I don't quite get the superior part.
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Why hasn't anyone mentioned sccy?
I agree with the suggestion above of the M&P Shield. I have a Shield in 9mm and like it. I didn't suggest it earlier because it's not a great range gun, it leans more toward the carry side of the question. But it's not a terrible range gun either and the price is right with the current rebate. Both because of the small frame size and because the OP is on a tight budget I'd suggest sticking to 9mm. It's both more comfortable to shoot and cheaper to buy than the other options.
I suggested a used Glock 19 above because I feel the 19 nicely balances range/target and cancealed carry concerns and it's at least in the neighborhood of the upper end of the price range. And if it needs service or parts they are very easy to come by. But I admit it might not be possible to find a fair condition used one for $300.
First Choice, Used Glock in any caliber 9mm or up
2nd Choice, Used S&W Model 10 or K-frame .357
3rd Choice, Used RIA or Norinco M1911A1 Clone
Lorcin or Raven, well for 300 why not both.
A little more than that:
"The Smith & Wesson K-38 Combat Masterpiece Revolver Model 15[1] is a derivative of the classic 1899 K-frame (medium frame) Military and Police .38 S&W Special (aka .38 Special) six-shot double-action revolver. The M&P underwent steady evolution throughout the 20th century and S&W spun off several variations as separate models in the post World War II years. One of these was the K-38 Target Masterpiece, which began production in 1947. The Target Masterpiece included a number of new and/or special features, including a six-inch barrel with a narrow rib to provide a level sight plane, a Patridge front sight, a micrometer click rear sight, S&W’s .375” short-throw hammer, a trigger adjustment for overtravel, and improved grips.[2] Noting the accuracy of the Target Masterpiece, a number of police departments and the FBI soon requested the same revolver with a four-inch barrel and a Baughman Quick Draw front sight. The result was the K-38 Combat Masterpiece. The major distinction between the K-38 Target Masterpiece and the K-38 Combat Masterpiece is the barrel length and the front sight"
It is superior AFAIAC, the people who used them to defend their lives for a living seemed to agree, and it certainly looks nicer. But, if you like more sturdy sights on the 10 then have at it. He also said he wants a range gun out of the deal.
If you are going for back then < $300...I'll take the model 28 Highway Patrolman I paid $150 for in 1975. Otherwise, today <$300 CZ 82.
Tanfoglio EAA witness 9mm polymer
Ruger P series, used