According to this page, the ず in 負けず嫌い is not negation, but from an old way of writing むず/んず.
It says むず connects to an imperfective form of verbs/some auxiliary verbs etc, and it's said that it's a transformation from
auxiliary verb "ん/む"+
verb "す" ("んとす/むとす"). It says "むず" usually has basically the same meaning as "ん/む", but it incorporates extra emphasis.
According to this page, it means something like 負けるだろうことが嫌い, or "hate thinking that you'll lose".
Alternatively, according to this page, it quotes the 新明解国語辞典 as saying 負けず嫌い is a mixture of 負け嫌い and 負けず, and says that it may originally have been 負け嫌い but came about via incorrect usage.
Daijisen says it might have come from a mixture of 負け嫌い and 負けじ魂 ("unyielding spirit") etc (I'm not sure which is correct).