Boy, $300 doesn't buy much in the way of a BC or BP/W these days (except, see below). Everyone is going to try to talk you into a BP/W and you MIGHT be able to get one for $300 if you don't need a stainless steel backplate but could settle for aluminum
HOG Custom BC Package You can change the plate later.
Could I ask why you want weight integration? More to the point, do you know why you want it? If so, skip this part.
In cold water where you need a significant amount of ballast (say 20# or more), carrying that much lead in the BC makes the unit, complete with tank, just too heavy to be any fun. Sure, you can wait to the very end to insert the weight pockets but sooner or later you're going to wonder if weight integration is such a good idea.
Perhaps a better idea is to use a weight belt or (better) a weight harness. Like
DUI Weight & Trim "Classic" Weight Harness System There are two interrelated goals: you can float on the surface in your wetsuit while wearing your weight system with no BC and, second, your BC can float on the surface with a full tank and any integrated weights without you and your wetsuit. You may decide that a good solution is partial weight integration. Perhaps split that 20# or so between the BC and the harness. Best of both worlds.
In the case of the BP/W and cold water diving, a stainless steel backplate makes up about 6# of your ballast. Perhaps the rest belongs in a harness. I don't have hips so a weightbelt is out of the question.
Now, for warm water diving where you only need a very few pounds of ballast, weight integration makes a lot of sense. I did a lot of diving where I only used 6# of weight and that was to account for the weight of the air in the tank.
You CAN get a back inflate, weight integrated, BC for < $300
Edge Stealth 2 BC I bought one of the obsolete, pre mod 2, versions for my grandson ($150, I think). It works pretty well and will serve his needs for a while but the more costly Stealth 2 is a lot better BC. Eventually, I will buy him a Deep Sea Supply BP/W to match the ones I bought for myself and my son-in-law. Maybe this summer...
If you haven't used a back-inflate BC or a BP/W and have only used a jacket style BC, you are in for a surprise. First, the front of your gear will be much more streamlined. Second, if you overinflate the bladder to get high out of the water like you could do with the jacket style, the excess flotation in the rear will tend to push you over. The trick is to inflate just enough to get comfortable. Of course, a lot of this will depend on the weight and buoyancy characteristics of the tank.
I would be remiss in not pointing out that you will be money ahead (eventually) to just get the BP/W now. Save up a little more money and do it once. Personally, I like the Deep Sea Supply products
https://www.deepseasupply.com/ but I would consider that HOG setup. The DSS with stainless backplate is only $470. So many choices...
Good luck!
Richard