I forgot to update what I finally did. I went with Cfair because they were so prompt and friendly with their communications, and assured me it was no problem to send money to a U.S. credit union without IBAN or BIC/SWIFT.
Registration was easy, transfer of funds was easy and the money arrived in the US account as expected. My Swiss bank didn't charge me anything, because it was a payment to another Swiss account. CF didn't charge me anything because I had a referral code. My U.S. credit union didn't charge me anything to receive the money.
All in all, I'm very satisfied.
I'm still considering a Schwab trading account (thanks for the tip fmf) as well, but for now Cfair worked quickly and flawlessly.