How about just using some hi-visibility reflective tape as a collar? That is passive as it uses the light of car headlights and street lights to make the wearer visible.
Why re-invent the wheel? This is what hi-vis reflective material is for.
If you really want to use LEDs then why not just have an LDR to detect when ambient light is low and then flash some superbiright LEDs. The superbright LED's can be used at very low current and will still be visible at night. The human eye is very sensitive to light in darkness; even more so if it's flashing.
As an example I use some blue SMD LEDs as power indicators on my PCB and run them at 5v via a 10k resistor. That's less than 0.2mA and is visible in daylight. You'd need a minimum of 3 LEDs for 360 deg visibility. 0.6mA or less, flashing 1:2 is an average of 0.2mA PLUS whatever the uC uses.
You could run an ATTiny at 1Mhz and it should use about 300uA in Active mode. That's 0.5mA total consumption.
Eking out the power is the biggest challenge, as I'm sure you'll want to use a small power source, like coin cells.
Something like an LR2450 is 120mAH 3.6V. That would run everything for approximately 240 hours or 30 days of 8 hours per day darkness.
It does mean the cat will be lit up all night, but you'll need to decide which is more important to you, the safety aspect or glowing cat.
You may well have all your neighbours reporting strange glowing alien beings in your neighbourhood for a while though.......