lcr meter have dedicated circuits or algorithm / calculations etc ... to do those measurements / esr ...
An multi-meter use the time constant RxC in capacitor measurements, witch is not enough for esr needs
it will be good to a point, it will never replace an good lcr who use frequencies / voltage bias etc ...
and a good example is an DE-5000 who's very effective for most current needs, even some cheap multi part tester can do some serious jobs
, you can find open source circuits or even the coding used to do theses jobs
an lcr meter is not an multi-meter and other way around, don't mix them
good luck finding / leaked fluke FW loll even the cheap uni-t models too, or even hacking them, you sound you want to get something but not hacking them or write FW for them loll
be realistic and grab yourself a real lcr, there's plenty available