RAVPower has specifically stated that they built their power banks to handle passthrough charging, so that’s a good option if you’re looking for a “known-working” product.
However, they also warn that using passthrough charging for “more than a few hours” will absolutely damage your power bank, so it’s a bit of a mixed message. The bottom line is that passthrough charging on any power bank causes it to heat up. The longer it’s in use, the hotter it gets, and the more damage to the battery. On high-quality power banks that will only decrease the battery’s capacity; on questionable power banks it could lead to physical damage or a fire hazard.
The only way around that is to use a purpose-built UPS which has the circuitry to quickly switch between power sources in the event of a voltage drop. Since miniature UPS devices often cost more than a basic UPS, the only reason not to go with a standard UPS is usually space, not cost.
There are also a number of boards that are designed to provide backup power for low-power USB devices like a Raspberry Pi. Again, cost (with battery) is about the same as a real UPS but if space is the issue it might be worth it: