Actually driving a 250k volume control from a low impedance will not affect its operation much at all. Passive basses use them all the time and the low output impedance of the op amp will just make it work a little better, not worse. The problems you describe would occur if you were trying to use the volume pot with a low impedance load. If you are intending to drive the normal 1Meg amplifier input impedance then a 250k volume pot will work just fine whether it is driven directly from the bass pickups or an opamp. If you intended to drive a low impedance load, headphones for example, then a 250k pot will not work well no matter what you drive it with. The effect you describe is real but it is caused by the load impedance, not the driving point impedance. The signal impedance is 1Meg in this case whereas the maximum series impedance the pot could introduce is 250k or a relatively small one quarter of the signal impedance.
In any event the second (bottom) approach does not have a volume control at all in passive mode and the active mode is unchanged from stock. It would be perfectly acceptable as a limp home mode and it is a very simple mod.
Oh, if anyone really cares the Reggie Hamilton circuit is not exactly like either one of those but it is more like the bottom one with one addition. Instead of passing straight through to the output the passive mode runs through a 250k volume pot that is ganged with the 25k active mode volume pot on the same shaft so that one knob controls both the active and passive volume settings. It is a nice approach but of course finding a two section 25k/250k pot for DIY mods is a little difficult. Some guitar parts places claim to sell "Reggie Hamilton" volume pots but I have never tried to order one so I don't know if anyone really stocks them. The complete Reggie Hamilton circuit is shown below.