Winchester has a compact version of the xpr in 243 and 350 legend. Both of those are soft shooting calibers. They retail about $500 I think and there is a $50 rebate if you buy before the end of the year. It's a bolt action, not single shot, but you could just load one round into it.
CVA Scout is available in a variety of calibers. Its a compact rifle but not sure if its available in a youth model? And if you can find them in stock they are listed for under $400.
My kids both started with a new england, 243 and 270. 243 was a youth and 270 wasn't. Had a friend take 2 1/2" off 270 and it was exactly same as 243. My nephew is looking and hoping to find similar. We've looked at all new guns. Hoping to find one set up and ready to go.
I just bought the Henry youth single shot.243 for my daughter for Christmas, seems to be a very well made gun and has really nice wood, I only paid 440.00 for it on Gunbroker brand new with free shipping. https://www.gunbroker.com/item/1081085246