here is a pic, of kr1 vs kr4, not mine pic, taken from google images, it makes size difference between kr1 and e10 insignificant, kr1 is actually longer by about 3\4 inch
The Fenix PD32 V2 (Osram CSLPM1.TG) W2 may be worth considering, as you like Fenix.
Locating the dog? Perhaps keep the flashlight you have: get a pet light? Buy Products Online from China Wholesalers at Aliexpress.com US, which in Pounds Sterling is ……hmmm, maybe even less?
With the left over money you can get a new battery and a new hat for your dog.
No need to yell at the messager if it’s not a good recommendation, just trying to help you out mate:-)
Never ever wear pumpkin hats like that, else his head will turn into bugs.
I got a similar fenix. side button had issues. did not feel like smooth press and had to really press hard 1/4 times. I exchanged it.
The new one was much better without issues but it doesn’t really have a ‘click’ type action.
One of the highest priced lights i’ve bought with the fewest features…. better be perfect 
Also i sold it because i like my other lights better