https://www.amazon.com/MTX-Terminator-TNE212D-200-Watt-Enclosure/dp/B001JECAM2/
That plus a budget friendly, but good quality monoblock amp would be my reccomendation for new subs in your price range. I have these in one of my vehicles that I didn't want to spend a lot on. For what they cost, I am happy with them.
Not sure what you decided to go with but IMO if you get a sub in your trunk then your speakers will sound like they're better. Many people on this subreddit will bash this recommendation but I'll give it anyway.
The converter is needed if your radio is stock. I also think it makes the install easier regardless as it is one set less wire to run & you won't have to take apart your headunit.
I think this is the best intro level set up for a subwoofer if you're looking to spend about $220. It will definitely impress you if you haven't heard/had bass in a car before. Also very easy to set up and you can do it in about 3 hours with no experience.
Edit: forgot this amp comes with a built in high/low converter to feed straight off the speakers. Converter may be easier but is optional.
Hey, sorry for just noticing your reply. The car doesn't have many wires as it is, but I am in the process of rewiring everything in it (new alternator, new gauges, new wires, lights, etc.) so I think the wiring won't be a problem - the car is so barebones I could fit everything wherever I wanted with little problem. As to the head unit, it was a random one from a likely Chinese company that I didn't find too offensive to put in my car, this is the link, I don't think it mentions what you were talking about.
As for the rest of the items I was looking at, after a quick look back at my history, these were the sub, 5 channel amp, and speakers I had my eye on. I am starting to realize the $300 I had in mind is sorta starting to sound absurd lol. Also, would the sub still be appreciated being in the trunk if I were to play music while driving?