Electronic – Replacing a Switch and Avoiding body diode conduction

I am trying to design a replacement for a 2:1 switch using FETs, which would be really easy if I didn't have to concern myself with the body diodes that are present on PMOS transistors.

My scenario is that I have two batteries in parallel to power a circuit, they are both fed into a dual schottky diode and then that output is used to power most of the circuit, this dual schottky is good as it prevents a major issue arising if one of the batteries were to short circuit – the other would drain through it and die quickly. However, unfortunately the voltage drop over the schottky is too high (even though it is a mere 0.25V) and would not be able to power a specific part of the circuit that I need, and it would stop functioning much earlier than if powered direct from the batteries.

I need to switch between the two batteries to power this part in order to keep their voltages relatively even. Originally I was using a TS5A3154 multiplexer to do this, however I ran into issues when I realised that the analogue inputs would be higher than the supply voltage for the chip (the enable pin became redundant and the output would always be high [whatever was at NC / NO]). So I want to replace this with FETs as mentioned (PMOS or whatever will work / be cheapest) and need to avoid the possibility of one of the batteries shorting allowing the other to drain through the body diode that is present on PMOS transistors. The reason I am struggling is because any solution can't have a voltage drop over it as this would eliminate the point of the switch in the first place.

The FETs (switch) is / will be controlled by a microprocessor which only has an output voltage of +2V, while the battery voltages will be up to 3V at the beginning of life and then will be considered dead at about ~2.5V as some of the circuit won't work and they are crucial for the functionality of the circuit. The premise of this being that a 2V pulse from the microcontroller will switch off / on one of the batteries and then switch on / off the other one accordingly.

Just to reiterate, the question is to prevent the body diode of a PMOS conducting given events mentioned above without causing a voltage drop when they are functioning.

Cost is quite a big thing with it needing to be as cheap as possible, I have searched for alternative switches to replace the current one but all datasheets seem to say the same sort of thing regarding analogue inputs being higher than the voltage supply.

I would draw a circuit to go with this question, however, I do not feel it would actually provide anything as all it would be is two batteries each going into a PMOS and then have the two outputs connected to each other going to some load. If you want one I will oblige but as I say, it will only give a small visual aid rather than any technical input.

Edit
Just to add a little bit more information regarding this question:

• The two batteries are required to power the rest of the circuit all the time while possibly only supplying this part of the circuit for part of the product lifetime (in reality it will power it for near enough the whole time but to make it clear, it can be turned on and off if needs be)
• The part I am wanting to power is for wireless communications, it will initially drain a few milli amps on power up but then reduce to somewhere in the micro amp region
• I can't just 'hope' that the batteries don't short, I need a contingency plan
• The rest of the circuitry will be drawing only ~20uA for most of the time of operation as low power is a must as well

EDIT PART TWO:

So firstly thanks to everyone who has given an answer already, I have looked into each of them but for one reason or another they aren't suitable for my application – largely cost, current draw, operating voltage, etc…
So to give you an update on my situation, I think the most likely way forward will be to use two TS5A3154 switches and have each battery power one each and also be connected to one of the analogue inputs on the switches and then somehow switch between the two using a micro controller (EFM32G222F128).
The new issue I am having is that at least one of the outputs from the micro controller will have to be always high, however I need my MCU to go into a low energy mode (EM2) but I don't know if it is possible to be able to do this while having an output high, so if anyone has any knowledge on this chip and coding it in C, that would be amazingly helpful!

Reference manual for coding the Gecko

For people who have already answered – I will still award the bounty if no answer for this addition is given so fear not!
Godspeed.