I wouldn't be embarrassed to ask for a refund on the difference in cost. In fact, I will be, although I have an iPhone 5s that I bought new, after the 7 models had come out. Despite the news saying this temporary price drop to replace the batteries only applies to the 6 and 7 models, I saw significant slow downs and shut-downs on my 5s after updating to iOS 10 versions. I have not updated to the 11 due to other reasons.
I began having these problems early this year, or maybe it was last year. However, I was out of town in March and was having major problems with my phone being unreliable and I need it when traveling. It was running slowly and shutting off at random, often at critical times. I was on the verge of buying a new iPhone SE when I decided to take it to an authorized Apple repair and retailer shop (there's no Apple store in that area) to ask what's wrong with it.
The tech ran a diagnostic on the phone and said, well you don't appear to have any apps that are causing these issues, and you don't allow them to run in the background or update in the background either, that's good. What we see is it's the battery - it's old so it's not charging up as well anymore. However, your phone is out of warranty at this point, so no getting a new one from Apple. From us it would cost $79 + sales tax to replace it, same as what the Apple store charges. I told him I'd think it over.
When I got home the following day, I went to my local Apple store to double check his findings. The tech there came to the same conclusion. Same price to replace the battery though slightly more due to the sales tax differences. No offer of trading in my phone right then for a discount on a new SE. I then said I'd think about it.
I talked to a friend who has the same phone and he told me I could replace the battery myself by buying a kit on ebay (which he had just done recently) for less than 1/4 the price Apple charges, or I could try any number of 3rd party shops in my city. Since I couldn't afford to be without my phone for a day, and I needed it to work properly all day every day, and I didn't trust myself to not mess it up by using a DIY kit, I opted for replacing the battery at a 3rd party shop for $35 + sales tax = $38.08, which is less than half the cost Apple wanted. The shop did it in an hour and it's worked great ever since. I saved myself from having to buy a new iPhone this year, which I really couldn't afford at the time, even though the SE is much cheaper than the newer models. If I'd taken my 5s to the Apple store and paid the $79 Apple charged, plus sales tax, it would have cost me $86.01 total.
The cost of the replacement battery the OP paid for probably includes sales tax.
I'm not holding my breath for a refund of the price difference, but I'm going to give it a try. Might start a lawsuit for people like myself and the OP, who feel Apple is unjustly gouging customers on the battery replacement cost, and hiding things from us.