A good lcr use dedicated precision lcr circuit, up to 100-200khz or even more
A Dmm or bench meter with say : capacitance measurement is a basic help tool, it wont check ESR D Q R etc ... I have Flukes 189, Gossen Metrawatt, BK Precision 5490, they all heck capacitor up to 30,000 50,000 uf, BUT they wont tell the ESR and many important tests factors.
A good dmm with theses added circuit may not be cost effective, if it was the case, we should have seen more of them ?
A good basic lcr would be an DE-5000, sure you have thoses small cheap lcr meter / component tester starting around 20$ who can do a good starting job.
If not satisfied, you could build your own meter / lcr and do a kickstarter design project ??
A de-5000 with is 2 dedicated / specialized chips, could easily be integrated in modern DMM, not sure if the dmm inputs could be shared in kelvin mode with the lcr section, you could lose some CAT specs doing so ? unless adding some input shielded relays $$, more input protections ... and relays over time can go bad and have contacts problems.