Ok, I picked a chip for each purpose and came up with this:
I used the LT1512 for the charger and the LTC4412 for the powerpath controller.
I'm assuming I'd use it with a 5 or 6 cell NiCd or NiMH battery.
The datasheet for the powerpath controller had SOME info for using it with a battery charger but I'm still not sure I "get" how it operates. It looks to me like there is SOME possibility for the charging circuit to end up powering the output, but I guess that would only happen if the aux source (wall wart) was not present and the charger was charging, which should never happen - because if the wall wart's not there, the charger won't work.
There are some pricey components out there for this sort of thing! The charger IC suggests a particular dual inductor that Mouser wants almost $7 for! I may switch the charger to the Maxim MAX712 because it looks like a cheaper build that way.
Still I'd appreciate comments from some of you who have done this sort of thing before I run off and build it.