In the kitchen of this old house are a couple of 2-tube 4 foot wraparound fluorescent fixtures with rapid start ballasts.  It's an old 2-wire system with no ground.  They have a hard time starting when turned on, since rapid start relies on a grounded metal reflector close to the tubes as a starting aid.  What would be the second best ballast choice for these fixtures? (the best being preheat, since it was in common use before grounding was common, but not easy to find now)

Some decent, better quality electronic.

Or connect the reflector via series combination of two "Y" rated EMI suppression capacitors (~2.2nF, bearing UL safety label related to it's "Y" rating) to the neutral wire - you make an effective ground for lamp striking, but it will stay safely isolated in terms of safety metrics.

These capacitor are used to make an RF short between primary and secondary side of switch-mode power supplies (so it cannot radiate using input and output cords as an antenna dipole), but ensure the leakage current to the secondary side is safe. Except for some newest types, they have to be used as a pair in series for class II devices (when the secondary is not grounded; to maintain safe isolation even when one of them break; those special types contain such two isolation barriers in one package, so using one such device would be sufficient, but you should be sure about such type)

Or if you can find one, a high power factor magnetic rapid start ballast is also good. Those seem to start more easily without a ground than the low power factor ones, plus the start up flicker is a bit more flashy as well.

Most of these are actually HPF ballasts.  They appear to be "recycled" commercial fixtures.  Previous owners of the house put them EVERYWHERE.

All of my high power factor rapid start ballasts start just fine ungrounded. I've set them up just lamps wired to the ballast and they still start fine. The only finicky ones I've seen are single tube low power factor ballasts. 

All of my high power factor rapid start ballasts start just fine ungrounded. I've set them up just lamps wired to the ballast and they still start fine. The only finicky ones I've seen are single tube low power factor ballasts. 

When lamps are kept clean and dry, they are not as sensitive on the grounding, but in the kitchen the dirt and moisture collect there very quickly, sowhy the grounding is so important.
And remember, thinner lamps are always harder to start then thicker, so where T12 worked fine, T5's might get into troubles (and consequently damage the ballast, if not protected properly).

In the kitchen of this old house are a couple of 2-tube 4 foot wraparound fluorescent fixtures with rapid start ballasts.  It's an old 2-wire system with no ground.  They have a hard time starting when turned on, since rapid start relies on a grounded metal reflector close to the tubes as a starting aid.  What would be the second best ballast choice for these fixtures? (the best being preheat, since it was in common use before grounding was common, but not easy to find now)

Ya can't beat good ole preheat!  All my fixtures will start whether grounded or not!  I have seen plenty of rapid starts, especially the newer ones approved for 34 watt lamps that will not start well unless grounded.  Most of the old school ones designed solely for F40 40 watt lamps will start fine with no ground. As mentioned, the single lamp rapid start ones are the worst.  The most awful starting fixtures I have ever seen are the 4 lamp rapid start F20T12 2x2' suspended ceiling fixtures.  Talk about a bonified POS!!!

There were enough issues with the wiring that I ended up rewiring all the ceiling lighting in the house, as well as outlets on those circuits.  Starting issues solved, though now I no longer live there.

Did you keep the fixtures / tubes when you moved away

I didn't.  Some of them got replaced with more household appropriate (incandescent fixtures).  Others were left where they were.  I did keep one or two tubes though, which I still have.  Tubes were nothing interesting.  Ones that stayed in use, or the two that I kept were SP41 Ecolux tubes.  Most of them were 25 watt UTSL tubes that had taken a beating being run on those higher output HPF ballasts.