Hello, I have to advise you against.
I drive a six-cylinder Z4 and recently also a 118i convertible.
I read a lot beforehand. I also advise you. Nobody does the work for you.
I bought the fully equipped car from a well-heeled 1st (senior) hand with actually only 10500km ..
After everything I've read before, the N43 engine (116i, 118i, 120i) is a problem engine. Keywords are timing chain (again and again), NOx sensor, injectors, ignition coils and a few little things more. Even if something has already been done on yours ... you say your budget is limited. But you should have plenty of money in your hand or leave it behind, or look for a ring pigeon like I was lucky to find. It doesn't necessarily have to be a BMW. And even I have made myself clear beforehand whether I am willing to invest if necessary. The car itself is great. The engine also appeals to me, from driving. But for special reasons only to drive with Superplus and every 2000km of injector cleaner additive in the tank. Not a problem financially for me, but for you?
So leave it. I've had almost 100 cars in my life and I'm not a screwdriver, but I am always extremely knowledgeable and also worked in the automotive sector for a while! I would think about another convertible or look extremely and maybe. find something better.
If it were a six-cylinder, things would look very different, but you won't find one with your budget.