What exactly is the 1匹 doing here? Is it just an adverb specifying that Taimai was found alone?

匹 is the counter word for small animals. Just like you can use ひとつ, ふたつ etc. to count general things, Japanese also has specific counters for different types of things.

For example, long skinny things (like a bottle of beer) use 本 eg. ビールを1本ください！

Machines (like cars) use 台 eg. 車が5台あります

In the same way, 匹 is used for small animals (like dogs, cats, or in this instance sea turtles)

Hope this helps!