It depends very much on you whether you need cross brake levers or are fine with levers at your bullhorn bars alone. Here are some questions to ask yourself:
Can you break effectively and quickly with your current brake levers?
Can you switch hand positions from tops to bullhorn ends in time for braking?
Would to be able to brake reliably with a set of levers at the tops?
Caveat: Cross levers may make braking from your current levers a little less controllable due to interrupting the cable outers. Which reduces the brake lines stiffness to compression and increases friction. However, this might not be noticeable in regular use.
Your Tektro brakes and levers are most likely compatible with any other product except Shimano. Bear in mind what pull your brake actually needs. There are short pull (road, canti) and long pull (V-brake) brakes.