All I can say is I have a daughther with --
two-syllable first name
one-syllable middle name
and
a long hyphenated last name, both of which are "unusual" to most American ears.
Everyone gets used to it. Daughter will probably opt for just one of the two hyphenated last names (I'm guessing the first) when she gets older and feels cranky about spelling 14 letters.
I will say that in the US, the "e" is going to be a challenge for most people. And in combination with the apostrophe, it might be too much for someone on the other end of the line (May I take your order? Can you spell that, please? Can you spell it again? Let me read it back to you? Oh, I got it wrong? Is that and "e" and then a space? Holy sh*t, let me call my manager..."

) Can you just use d'Angieri-Dharmapuri?