Hello all, new here.

I just bought a pair of Tektro TL720s (https://www.tektro.com/_english/01_pr...r&sort=1&fid=3), and was dumb enough not to notice it is for internal routing through handlebars.

I have aluminium bullhorns without holes, what are my alternatives?

I've considered drilling holes, although I'm not familiar with where and how I should go about doing that, as well as the safety issues.

Is there any way to have the cables on the exterior of the handlebars?

If the horns are reasonably stiff (I.E. not junk), you'll be fine drilling them. The bars that came on my Felt have a 2.0cm slot cut on the bottom about that same distance away from the stem. If you don't have access to a mill, find a good bit slightly larger than the housing and drill 3-4 holes in a line. Use a file or dremel to create the slot and dress the edge. Once you wrap the bars, the tape should hide any unevenness. I'd wrap the bars in painters tape and use a rag when clamping them to avoid scratches.

Alternatively, swap them for something like the RL720, which would be allow you to run the cable externally.