How hot do your Lionel lighted number boards get?

If they're getting that hot, I'd be concerned as well.  You might consider doing something to drop the voltage to the bulbs, you don't need that much light to light up the number boards.  Can you wire the bulbs in series?  Failing that, I'd probably look for some lower current replacements or even LED's.

I have another locomotive with lighted number boards, the 38079 GS-2 Daylight. Those ones stay at room temperature, and the harness it uses is rated at 18V 80MA. Might be the one to go with?

If those other bulbs are lower voltage, I'd try this one.

I'm curious, the rewired harness has a totally different connector, what's up with that?

Those are just the part pictures off Lionel's website. The advice I got from them was to cut the bulbs from the incorrect connector and splice them into the original harness. I took extra care to keep the wiring configuration exactly as it was before but they still got very hot.

I'll see if I can get them to just send me the other one in exchange for the incorrect ones they advised me on. I can't stand that $9 shipping charge every time...

I'd just use the 18V bulbs, they won't run as hot. 

Does the MA rating make that big of a difference as well? Was combing through Lionel's parts page and found one that might even be better, from the 38029 UP 9000. Rated at 18V 50MA:

Yep, the lower the current rating, the less power it'll dissipate.  Virtually all the power to an incandescent bulb is released as heat, the more power it consumes, the hotter it gets.  Also, a bulb rated at 18 volts and running on 12 volts will run at a lower current (and power dissipation) than the higher rated bulb.  That's the best of the choices so far.