How can a linear circuit with a pulse become non-linear when measured?
When it is clipped by the amplifier.
I suspect you want to measure battery ESR vs SOC using pulse charging perhaps with the flyback method or rectified currents with high dV/dt limited by ESR.
Do you know how to measure this?
Then measure all aspects of the pulse. Peak/rms {V,I} ,Pavg, duty cycle and correlate with DSO to choose best test method to create with ADC.
(1) Measure Vdc and store results for a few sec with 1A CC load and a look-up table for % SOC
(2) Pulse load ΔmV/1A = ESR [mΩ] for last 2 values.
(3) Measure all with DMM and DSO and examine critical inputs and outputs, d.f. peak, rms , C rate, life aging effects on ESR, Wh capacity for each charge for research using only 60% SOC each cycle or whatever.
(4) Make a list of goals. Compare charge methods for efficient, effective lifespan Wh delivery from SOC before after or total Coulombs to charge X % SOC.
Why not try your DMM in both AC rms and DC and compare with a DSO the report results above with a question? Various methods prefer to use a DSO as DMMs tend to be low BW. (<20kHz) but if your currents are step voltages, you want to determine Vavg DC and also Vrms AC for experimenting with Pchg or Irms AC charge rates. for 1C or less but you may also want 10C pk for good batteries 1us to restore some life but not add heat.
Consider this IC for current sense.
https://static.3peak.com/res/doc/ds/Datasheet_TPA191.pdf
If you wish to measure bipolar currents then Vr = Vref/2 for an offset input.
RMS applies to V or I as the average of N rms cycles for some time interval. So fin and fADC need to be known to prevent flicker.