Brundle have a few off the peg ones and will do custom ones. I have no idea if there is one near you I'm afraid but I've used them quite a bit (not for these though) and they've always been quick and helpful.
I'm planning on doing it in the future. I've got an artist friend who will take the photo of the customers pet and draw onto some ply board so I use that as a template for the plasma cutter. Like you my hands are not steady enough to do it free hand.
It is a very easy shape, why don't try to do it with a jigsaw. It'll be nosey but you should not have much trouble.
You can get great results from burning around a template just measure the tip dia. and half the size you find. e.g my tips are 9mm so I reduce the template 4.5mm all the way around.
You will see on the example that I have posted if the part you are going to make has holes in it bore a 3mm hole where they are to be on the template (I use 3mm aluminium ply burns) then transfer them onto the plate you intend on burning,then punch or bore at a later date after you have cut the desired shape.
I clamp the template to the material using Clecos/skin pins because if you have to take the template off for any reason it goes back in the exact position that it was in.