I think that you are expecting a bit much of a mechanical relay for the small time intervals that you require. As hevans said "Solenoids (and mechanical relay contacts) are relatively slow things".
My suggestion would be to build an electronic circuit with a timer(555?) and a Mosfet.
Martin, Martin, Martin, I'm afraid you still haven't got it.
"12v.psu, diode Vf 3.2v, @350ma . So, 12 - 3.2 = 8.8.. So 8.8 / 3.2 = 2.75 ohms??"
Why did you divide by 3.2?
R = (Vs - Vf) / If , not (Vs - Vf) / Vf
There is usually an increase in headlight brightness when relays are fitted. Some cars will have a marked increase, while others will have a smaller increase. Depends on the car's wiring system, and the condition of the wiring.
Fitting relays is the first step to take when trying to improve the...
R = (Vs - Vf) / If
So your first example would be;
R = (12 - 2) / 0.02
= 10 / 0.02
= 500Ω
For multiple LED's in series, Vf is multiplied by the number of diodes.
Vf for your second example would be; 3 * 3.4 = 10.2
R = (12 - 10.2) / 0.02
= 1.8 / 0.02
= 90Ω
The voltage in a car is a nominal 12volts, but varies quite a lot during normal operation.
Voltage can drop to 10v, or less, during cranking and then up to around 14v when the engine is running.
Automotive accessories are made to take these varying voltages without failing.
The ioniser will...
Do your ones require you to switch them on at night and off in the morning?
The only outdoor LED lights I have seen are either;
1- Solar cell on top charges the internal battery during the day and the battery powers the LED at night,
2- Connected to an external mains transformer.
Probably just wired the headlights through a relay.
Headlights will be brighter because they will be running at full battery voltage, not a reduced voltage through the car wiring.
As above plus,
What type of bulbs are they?
Are they all the same type?
Is it only one type of bulb causing problems?
Are your neighbours having the same problem?
Are you in a stand alone house or block of flats/apartments?