First, you should post the text here of the two emails you received from United. There's speculation going on about the first one. If it was the usual email that says your reservation is confirmed, you don't have to do anything, and you'll get a followup once they've completed 'internal processing' then you're going to be fine 100% for sure on this. I just don't remember whether there's something different than that on revenue tickets where certain segments are operated by partners.
Second, you should escalate with United. They took a credit card on the booking. Their screwup, or that of their alliance partner, shouldn't be your concern. At the very least it shouldn't cost you extra.
For those wondering what the actual DOT rule is, for the purpose of the Department's ban on post-purchase price increases a 'purchase' is deemed to have occurred when the full amount due has been paid. Presumably, and I haven't investigated this point, that this would be when the credit card has actually been charged and not merely when the credit card information has been taken. (That's how it works in other legal contexts I'm aware of.)
"§ 399.88 Prohibition on post-purchase
price increase.
(a) It is an unfair and deceptive
practice within the meaning of 49 U.S.C.
41712 for any seller of scheduled air
transportation within, to or from the
United States, or of a tour (i.e., a
combination of air transportation and
ground or cruise accommodations), or
tour component (e.g., a hotel stay) that
includes scheduled air transportation
within, to or from the United States, to
increase the price of that air
transportation, tour or tour component
to a consumer, including but not limited
to an increase in the price of the seat,
an increase in the price for the carriage
of passenger baggage, or an increase in
an applicable fuel surcharge, after the
air transportation has been purchased
by the consumer, except in the case of
an increase in a government-imposed
tax or fee. A purchase is deemed to have
occurred when the full amount agreed
upon has been paid by the consumer. "
DOT notwithstanding, and you can of course file a DOT complaint -- there's also small claims court. Remember there may well be a promise by United documented in the email.