The resistance of a cold tungsten filament is very low so the initial current is huge, easily enough to cause the loss of some volts along the wiring and particularly across any connectors that aren’t as tight or clean as they should be. Always worth a check to see if any connectors are showing signs of getting hot.
Don’t forget to check the path on the “cold” side of the lights/instrument panel too, if the earthing is bad the effect is the same… fewer volts where you need them.
If the wiring and connectors are in decent condition and you still have the problem then you can run in a second wire (fused!) to supply the lights via relays switched by the existing wiring.
When you say you measured the battery voltage, was that directly across the battery terminals? If there is a significant drop there then a new battery may help, though normally the first sign of a dying one is the starter motor struggling to spin the engine.