Very simple to check ....and you don't have to guess.
Hook a volt meter to your battery ground post and measure your open circuit voltage at the battery for reference, it should be @ 12.5-12.6. So now you know what voltage is available to your lighting.
( the video shows 12.0 VDC as a reference only and as stated your open circuit voltage will be @ 12.6 with a fully charged battery)
Take your voltmeter positive lead and measure voltage as explained in this video.
With your problem lighting circuit turned ON
...backprobe and read your positive + voltage going in, it should and be very close to your battery reference voltage a max of .5 or less of your battery reference voltage.
Now take your same positive voltmeter probe ( and with the lighting circuit turned on
) backprobe and measure the voltage at your ground side ....the voltage should read less than .5 VDC. If you read anything higher than .5 VDC on the ground side you have resistance problem on the ground side of your circuit and you need to find it and repair it ....or as described run an alternative ground wire to the problem circuit and IMO if you have a ground problem your best bet is to run an alternate ground wire and be done with it.