Yes they will play up to 85dB with 1 watt. It's really, really hard to know if that will be loud enough for your taste if you are not familiar with how loud a given dB level sounds, but 82dB is the average listening level in a mastering studio. Peaks may go way louder than that for an instant.
I don't have any experience with the 1W amp you mention with headphones or speakers. The last Berning amp I heard was about 20 yeas ago and I think it was 10W. Sounded great by the way. If you have tried the 1W amp with 87 or 89 dB speakers and liked it I think it will work fine for you with the Pipettes. I have used them with a 2 watt S.E.X. amp and it works just fine. If you have tried the 1W amp with 104dB horns and liked it it's really difficult to say if it will drive the Pipettes loud enough for your taste. These are intended to be near field/small room/apartment/desktop speakers. Though I will say they are doing a pretty nice job in my 20'x25' room with a small subwoofer.
