Switching power sources with a FET while avoiding brownouts
It makes no sense to say that diodes are out of the question.  Two Schottky diodes seem like exactly what you want, although they should be inside the regulator feedback path so that their voltage drop is compensated for automatically.

Another option is to make this "power bank" (whatever that really is) put out a bit above 5 V, and then have a power supply that puts out even more.  Both these feed the input of a buck switcher that makes the regulated 5 V for the computer.

The buck switcher will automatically adjust to whatever input voltage it is given.  With the power supply having the higher voltage, like 12 V for example, and this "power bank" maybe 9 V, the power supply will always be providing the power when it is on.

Added after "power bank" defined

You now say this "power bank" is something you made that contains batteries and a regulator to keep the output steady.  You should fix your question to define this.  "Power bank" by itself is ambiguous.

The conceptually simple answer is to have the power supply and the power bank feed the input to a 5 V buck switcher, each thru its own Schottky diode.  Adjust the power bank voltage to be a little above 5 V after the diode, like maybe 6 V.  Now the power supply only needs to result in higher voltage after its diode than the power bank does, and the power will be taken from it instead of the power bank.  With a buck regulator, there is a wide range of voltage this power supply can have.  Just about any 5 V buck regulator you can find will work with at least 12 V in.  That's a common power supply voltage, so there will be many readily available alternatives.  9 V will also work fine, for example, assuming the power pack puts out less than that.

A better approach is to step back and look at the whole problem.  You want to run your 5 V device from a power supply when available, but from batteries when not.  Therefore build one device that does that.  Only a single switching regulator is needed, unlike the two with one in the battery bank and the other to make the final 5 V.  Use a single 5 V buck switcher, with diodes to get its input directly from the batteries or the power supply, whichever has the higher voltage.

Put the batteries in series so that the lowest voltage is always above the desired 5 V output.  Choose a power supply with higher voltage than the highest battery pack voltage.  Use a microcontroller to shut down the batteries when they get too low.

For extra credit, charge the batteries when the power supply voltage is present.  This again would be managed by the micro, which would also do the charge balancing.

Basically, this will be a 5 V UPS.