Efficiency when drawing a mere 300 lines is probably not something you need to worry too much about. Even the old OpenGL screensavers available with Windows 95 and which used Microsoft's software implementation were able to handle that kind of workload with ease, so I suggest that looking for a more efficient method of doing something that's already an incredibly lightweight task falls squarely into the realm of micro-optimization.
If glBegin/glEnd works fine for you in this use case, then by all means continue using it. Yes, be aware that it's not the best way of drawing stuff, but you're already going to have sub-millisecond frametimes, so why bother with a program like this? If you're already getting over 1000fps then being able to boost that to 2000fps really won't mean much at all. This isn't a performance-critical program, 10fps would be fine, 1fps would be fine. You've better things to be spending your time doing.
Remember - Quake used glBegin/glEnd back in 1996, had much heavier workloads than a mere 300 lines, and ran quite well despite that.
More important things here are that you don't accidentally fall back to software emulation, and that you don't chew through too much power while running this all day.
You can avoid software emulation by either drawing quads (if you need wide lines) or by strictly restricting yourself to width-1 lines (which are more likely to be hardware-accelerated).
For power-saving, this is one case where using Sleep in your main loop is appropriate (don't even think about using it to control framerate though; it's not for that, it's for power-saving). So by all means go stick a great big Sleep call in there; you're dropping alpha by 1/255 every four minutes so Sleeping for 15 or even 30 milliseconds between frames is perfectly alright here.
This is all in the spirit of fitting a solution to the problem, rather than trying to force the problem to meet a pre-concieved idea of what the solution should be, and the approach you've currently got is fine for the problem as you describe it.