By "bike," you mean motor or pedal?? In either case, not a problem if:
-- have the proper customs papers for the 2-wheeler;
-- and a Lao visa in your passport.
It's about one km from Huey Khon to the Lao border gate, don't ask me why. Insurance place is right there after the gate on the left. It's another few meters to Lao Immigration and Customs in a building up on your right.
Two routes on to Luang Prabang:
-- more common and longer is from Hong Sa SE down to Sayaboury, then north to the Mekong crossing at Thaduea-M. Nan, then on to LP;
-- other, shorter and reportedly much rougher, is straight on from Hong Sa and ENE to Xieng Mene crossing and LP. Best to ask around there at Hong Sa as this soon after the rains might be a tad difficult. Haven't seen any recent reports here.