Choosing a well filter for household use
Your cost restriction may not be easily met with my recommendation, but it's close. The DIY portion also requires that one be capable of managing PVC plumbing connections, which isn't particularly difficult.
I have a chlorine injection water treatment system that the local plumbing supply house would have installed for US$3000, but sold me the components for about US$1800.

The above image from Clean Water Store shows the routing of pipes and components necessary to assemble such a system. The diagram is slightly different from my installation, only in that the injection port is between the pressure tank and the settling tank. It's called a contact tank in the image.
The incoming water has chlorine injected from the &quot;Stenner pump&quot; which may be a brand name. Not shown in the image is a flow switch, which engages the pump when water is being used by the house. The injection volume is adjustable. Chlorine and water is in the solution tank.
The combined chlorine and water then &quot;rests&quot; in the contact tank. Some of the iron (most of the iron) will react with the chlorine (rust never sleeps) and settles to the bottom. The remaining iron will still be dissolved in the water.
The chlorine is removed by the carbon filter. One adjusts the chlorine injection so that the water exiting the carbon filter has no chlorine level. Pool test strips are useful for this testing.
Not shown in the diagram but present in my system is the water softener/conditioner, which will additionally remove some iron.
I have a pair of cartridge filters after the softener which are installed as pure white and when replaced every three months are iron-brown.
I can occasionally taste a bit of iron, but rarely is there any hydrogen sulfide smell, unless there's a failure somewhere in the system. It's necessary to check the solution tank periodically and refill it. The injection valve frequently collects iron deposits and has to be replaced. Both expenses fall into your US$250 limitation, although it was necessary to purchase the injection valve &quot;duckbill&quot; online to get a decent price. I believe it was six or seven dollars online compared to nearly thirty at the local supply house. I replace them every four months to avoid the nuisance of waiting for a failure.