In one of my classes this year, there is a LED tube in a fixture with a programmed start ballast. In case anyone's ever wondered how they start, it basically is like a little rapid start. When the cathodes are heating it glows pretty dimly and then comes to normal brightness when the other lamps turn on. It's a WW LED tube so that's nice, I 100% support a 3000k-ication.

Also would the LED's shorting caps hurt the ballast during the heating stage?

It depends on the exact ballast schematics. Most of the time it is okay, as programmed start ballasts (electronic) usually have some kind of current limiting elements in the filament heating network. But not always. We had an example here when a badly designed ballast (GE, IIRC) had PC board traces blown when connected to a couple of LED tubes having the ends dead shorted.

LED tube manufacturers sometimes keep a list of compatible ballasts, but honestly, the whole idea of running LED tubes from electronic fluorescent ballasts is a quackery from the start... Too many possible combinations, too many weird effects to happen.

I have seen an LED recessed can trim that was connected to a programmed-start ballast in a CFL can, and it started just as you describe.

One time, I put these LED tubes in a fixture with this ballast.  It did exactly what Emersyn said; dim and flickering during the preheating stage than coming on at full brightness.  I left it that way a few months before ultimately restoring the whole thing back to T12 fluorescent.