A good suggestion.
Though I tried a "led replacement" for my dome years ago (what a joke!), the newer LED is great. MUCH higher & better output. It seem to illuminate much more - maybe its much whiter light - and it isn't wrecking its holder & cover like the old 5W bulb's heat did. And its much brighter than a 10W bulb.
IMO no doubt - LEDs are now the superior lighting method.
Using LEDs with or instead of the dome etc bulbs assumes PWM dimming if linear dimming is desired.
If the LEDs dim linearly, it's PWM - ie, LEDs should dim fairly proportionately to the bulbs.
If analog, the LEDs will tend to dim slowly from full bright before a sudden dive to off (probably when bulbs are still ~1/2 bright).
Standard DMM voltage ranges cannot determine PWM signals, though some DMMs have a frequency setting that might be useful (assuming you aren't measuring noise - but PWM frequencies are constant except for rare situations).
A True-RMS voltmeter should detect PWM. With a load (resistor, LED, bulb) the RMS voltage should drop. The RMS voltage (like the current) should reduce linearly (in proportion) with the PWM duty cycle.
Ironically (pun), old moving-coil voltmeters meters should also indicate PWM as their moving iron mechanism (get the pun now?) measures the average voltage & current.
Though DMMs are supposed to measure average values, they actually measure a peak value and scale from that - ie, x1 for (ripple free) DC, and x0.636 for AC. (ie - "sample & hold" circuits: sample, hold, then convert to digital; then re-sample & hold etc.)
FYI - For a sinusoid, the average voltage = 0.636 times the peak voltage. Same for current. Hence incorrect readings for non-sinusoid waveforms. (Big impact on current readings for SMPS systems!)
BTW - signal frequency can have an effect. But True-RMS meters should have a wide bandwidth - ie, measure from DC to many Hertz (100kHz? MHz?) so that all sinusoids are measured. (Any wave shape can be represented by a series of sinewaves at different frequencies. That includes square waves, triangular waves, music waves, though I'm unsure about Mexican waves.) I assume True-RWM meters (still?) use DSP techiques (Digital Signal Processing).