Sorry for the confusion. They don't object to me having a US passport and adding Czech citizenship. In the same way, I'm sure Britain doesn't mind you adding US citizenship. But the countries, in my case Czech, in yours the US, that are being applied for, do mind.

The US Embassy hinted that I can sign saying that I've renounced my citizenship, even though I haven't actually renounced it. I don't know if that works in your case, however.

It's more nuanced yhan that. You have to renounce your citizenshop, but after Britain doesn't recognize that you'll be a dual national. See: http://www.richw.org/dualcit/faq.html