Flyte and Barker was a broomstick manufacturing company. The manufacturers of the Twigger 90 broom, which was developed in 1990 to rival broomsticks from the Nimbus Racing Broom Company. Although expensive, it had an unfortunate tendency to warp.
Flight and Barker is posibly play on words on flight embarker, or someone who embarks on a flight. Flyte is obviously a play on the word "flight", while "barker" refers to the bark on a tree, which talks about the materials that a broomstick is made from. It is also possibly a play on the phrase "barking mad", since the Twigger 90 was thought to be a broom for those "with more Galleons than sense." It may also be that there is some element of the carnival "barker", whose job was to talk up the attractions of a fair or circus and incite the public to pay to see them.