The more 'electronic' a power supply, the longer it takes to start up. I had this nuisance in my pantry, where I only need light for a few seconds, so I don't want to wait another few seconds every time. I also have a VERY high current demand from a lot of LEDs in total.
I used a sealed lead acid battery on a trickle charger, and a relays. Battery is on common. NO is connected to LEDs. NC to a trickle charger.
Yeah. I'll have to replace the battery when it goes out. Beats waiting for a power supply to ramp up.
Long term, I think I'm going to make a service grade (100-400A) 12/24/48v DC bus for the house, with capacitors for buffering.
You might try a very simple 120vAC to 12vAC transformer (assuming thats your LED's voltage) with a rectifier and capacitor in some enclosure. You're building electrics at that point, so test, and include fuses, etc.
There are fast-start power supplies, and those which are less fast. The latter tend to be less expensive as well so paying more might get you faster response. Or might not.
I have an LED-equipped light fixture in my living room, about 300W-incandescent equivalent, which starts instantly and is dimmable. It cost me about $300 just to put those bulbs into it.
But yours is what it is, and the only way you'll improve it is to replace it -- which means finding a way to return it, or selling it off to someone else who's willing to tolerate the pause, or learning to tolerate it yourself. Wish I could tell you otherwise, but...