adding bullhorn handlebars, what about shifter and rear derailleur?
We don't do product rec here, but I will note that SRAM rear derailleurs are compatible across road and mountain groups for 9 speed, as are Shimano. 

So, if you have a Shimano rear derailleur, buy another Shimano shifter and keep the existing derailleur (or Shimano compatible, e.g. from Microshift). If you have a SRAM rear derailleur, replace Shimano with SRAM in the last sentence and ignore the parenthesis.