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Diodes are not needed since the voltage will stabilize at the voltage of the battery with the lowest voltage then there will be no more discharge between batteries. If one battery is +9.1V and the other battery is +9.0V then their paralleled voltage will become +9.0V.Diode isolate the two battery packs to prevent self discharge. It will cost you 0.7 volts at the input terminal of your device.
I connected Ni-Cad batteries in parallel for my beach sound system many years ago. It worked perfectly for a very long time.If you don’t believe me, try it. Get a couple of rechargeable batteries and charge them to the same point and note how long they take to discharge. Recharge them and connect them in parallel and monitor the discharge time again. To be thorough, recharge again and monitor the discharge in isolation again to verify that the batteries’ discharge characteristic has not changed from the first test run.