No. Your statement is unclear. What do you mean by “could”? Natural things to say as the buyer: The asking price was $550 but I got him down to $520. The initial price was $550 but I got it for $520. He wanted $550 but I got him to take $520. I bargained him down from $550 to $520. I’ll give you $520 for it. He’s asking $550 but I’m hoping he’ll take $520. The stated price is $550 but I won’t buy it if I have to pay more than $520. He wants $550 but I’m hoping he’ll take $520. As the seller I’ll take $520. I’ll reduce the price to $520. I wanted $550 but I’ll give it to you for $520. I got $520. You wouldn’t say (at least in the US) that you were bargaining, haggling or bartering to the person you’re doing business with. You would just do it. You could describe what you did to someone else using those words.

If you are the seller, you should say, "The initial price was $550 but I could REDUCE (or mark it) it down to $520". If you are the buyer, you could say, "The initial price was $550 but I would like to HAGGLE and get the price down to $520" or "The initial price was $550 but I would like to BARTER and get the price down to $520". To say, "bargain it down" sounds unnatural. A person can get a bargain or receive a bargain when the price has already been lowered. If a person talks to the seller and negotiates a lower price, English speakers would use the expressions, negotiate the price, haggle or barter. If you are good at getting a seller to lower his price by negotiating, you would be described as "driving a hard bargain" or "clinching a good deal". Hope this helps

Yes, grammatically it's fine but shouldn't the bargained down price be lower?

Yep, that's fine.