This is essentially my circuit that SHOULD get 9 v from the barrel jack connector and if that cuts off, draw the current from the battery. Basically the beginning is a simple battery backup system I found, but I find it to be rather silly since it draws current also from the battery draining it and whilst it can be used (with a resistor) to charge a battery, it doesn't have overcharge protection, so before redesign that bit, the beginning of the circuit is essentially a placeholder.
Essentially the next bit uses an opamp as a buffer to create a virtual ground to be used for the main circuit and then two 500 mA linear voltage regulators to regulate the post-diode ± 4.2 V currents to ± 3.3 V currents.
The datasheets for the regulators are:
https://www.analog.com/media/en/technical-documentation/data-sheets/ADP1715_1716.pdf (for the positive rail)
https://www.analog.com/media/en/technical-documentation/data-sheets/ADP7185.pdf (for the negative rail)
What this circuit will eventually power is a microcontroller (connected to the +3.3v rail and virtual ground) and several analog sensors (for which I need the negative rail) and feed the result after rectification to the microcontroller.
Is there some huge fails in there I have missed?
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Update:
From the great comments I modified the circuit further, dropping the battery backup since it's not really needed at this stage. I went along with the BJT transistor suggestion, since I thought it would be the more cost-effective option than a specific opamp (the versions of the jellybean transistors I chose should be able to carry 600 mA according to their datasheets).
I mean... how hard can it be to create a regulated ± 3.3 V from a single 9 V source?
I also considered using REC5-1205DRW/H4/A/SMD (https://recom-power.com/pdf/Econoline/REC5-RW.pdf) as a costly but straightforward alternative given that it's also been rated for medical devices, but it would probably be a "tad" overblown and it too would require also a negative source voltage and to acquire that I would need to drive the bastard with 18-36 volts in total, which I think would be really overdoing it.