Why not simply use a word equivalent for ‘A’, such as: 哆啦欸梦
IMO that's because 欸 is rarely used in Chinese writings, despite its very common use in spoken language. I've never seen any foreign word with /eɪ/ sound translated into 欸. AIDS(/eɪdz/) is translated to 艾滋 or 爱滋, not 欸滋, even though the latter would be closer to its original pronunciation.
Refer to this ngram chart. 欸 is simply not found in Ngrams, while 诶 is so rarely used compared to 阿(which is used a lot in the translation of foreign words, e.g. 阿拉伯、阿莫西林 etc.) and character A itself.
Is this an exception or common among foreign loanwords?
Yes and nos. There are a lot of words borrowed from foreign language retained a foreign alphabet in their spelling. E.g. X光(X-ray), T恤(T-shirt), B超(B-scan ultrasonography, or B-scan), 三K党(Ku Klux Klan, commonly called the KKK). What is common to them is that the alphabet is left as-is during translation. However, words like 哆啦A梦 or 卡拉OK(karaoke) is much rarer in that, the alphabet is a result of no suitable character to correspond to that sound.
At the end of 《现代汉语词典》, there is a list of words that are considered Chinese but begin with an alphabet (西文字母开头的词语). Other words with an alphabet in the middle can also be found throughout the dictionary.