In order to know how to understand it here, you have to understand the function of the words that are being used in the first place.
「残念」 here acts as an adjective. How can you figure it out? Well you have to ask yourself those questions:
1) How is it?
2) What is it?
And try to answer them yourself. You will see that the answer to the first question is an adjective, and the answer to the second question is a noun.
So in your sentence, if we assume that 「残念」 acts as a noun, then the following 「で」 is without doubt the 連用形 of the 助動詞 (auxiliary verb) 「だ」. As for 「も」 you already know the meaning.
So we could interpret it as following:
残念でもしかたない。 : Even if/though it is a regret, it can't be helped.
However, if we assume that 「残念」acts as an adjective, which is the most likely, then 「で」 is the 連用形 of what we call a 活用語尾 (conjugative suffix).
Also, an adjective is a qualifier which is used to describe something, so the way you translate a noun and an adjective will obviously be different.
Anyway, you could interpret your sentence as following:
残念でもしかたない。 : Even if/though it is unfortunate, it can't be helped.
The difference may be small, but you can clearly notice how the second interpretation makes much more sense.