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.
parallel a LED (string) with the dome.
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).