Electrical – Is high impedance important for a differential amp measuring a battery voltage

Using a resistive differential divider as the input to a differential amplifier with a lower voltage rail than the Vin being measured is valid BUT be aware (and some simple calculations will show you) that the common mode voltage does not cooperate properly with the division and you get increased error.

AFAIR error effects due to errors in the divider resistors act on the full common mode voltage whereas the dividers themselves divide only the differential signal. So any errors in the divide resistors effectively increases by the error % of the resistors. If your common mode voltage is large compared to th edifferential signal "there will be problems".

Very approximately if you have 20:1 divide and 1% resistors you can get ~= 20% errors. Using 0.1% resistors as @The Photon suggests may seem a luxury but isn't and will probably only deliver accuracy about as good as you have said is OK if unadjusted. Note that this error may be removed by either physical calibration or software adjustment and you are then just left with the time variant effects. eg a 25 ppm/K resistor may end up looking like 500 ppm/k or perhaps 1% error at 20 K variation.

All the ab0ve is approximate as you have to see how combining worst case pars works in the circuit you use, and eg temperature effects will usually track and so partially cancel to a large extent.

If you can provide a floating high side supply you can buy isolation amps that have specs better than you need.

Here are Digikeys reasonably wide range of offerings

You can use these elegantly clever dual linear optocouplers
to form an isolated linear interface.
This is Vishay IL300 but there are others.
The circuit relies on the balance in the two receive diodes- ne being used on the send side for feedback linearisation and its (hopefully) good-twin on the receive side for output.

In simplest form

Usually done more like this, but still simple enough

A TL082 is somewhat inappropriate for low-level instrumentation- the offset can be as bad as 20mV and the drift is not guaranteed but typically is 10uV/°C. A gain of only 1000 means that a 10mV offset will saturate an amplifier with +/-10V supplies, even with 0V in.

If it was 1975 and you really had to use something that bad, you could put an AC voltage on the bridge and measure the AC voltage coming out of some AC-coupled amplifiers.

However, it's not 1975 and we have access to chopper-stabilized "zero drift" amplifiers that have offsets in the low uV and drifts measured in tens of nV/°C.

As an alternative, and since you're using a microcontroller, you could use the micro to alternate the voltage across the bridge and do synchronous detection, which is more like the first method, but involves a bit of firmware. Input noise voltage is 25nV/rt-Hz in the white noise region but you may not be able to achieve that, depending on the 1/f corner frequency.

Depending on how you want to proceed, you should pick one of the above methods. It's not entirely clear from your question whether you're trying to measure to a precision of 20uV or whether your entire signal is 20uV and you wish to measure it to a much higher precision. I assume you're actually looking for a resolution of 20uV, which is not that difficult to achieve in practice.