Making a companion is not as difficult as it seems in fact is actually really simple is just very time consuming
I see you already have all the star and blender so you're already on the right foot. I actually did it tutorial for this process a while ago but this was when I had first started doing Bento rigging, so it's not the best method and that video and I definitely need to update it.
However that doesn't mean that I can't give you a rundown of the process.
When I make bento animals I use blender 2.79 and Avastar 2.79.
So let's begin okay
The first thing you need to do with the import your animal that you want to be I know you said in your post that you want to be a bunny but I'm going to use a dog as an example you can still do the same process it's just a lot easier for me to explain it using a dog because I am recalling it from memory.
So let's say you had your dog put it onto your first layer , you're going to create a new layer and import of avastar body into blender.
For this we're going to use the extended body. Technically we only need the skeleton but I found that it's a lot easier if you have the human body on because for some reason if you disable it you have problems with weight painting later on I might be wrong. I'm so just having it there for humans that is just you know habit.
So we're going to select the skeleton in object mode and move it to its own layer leaving the fleshy human parts somewhere else.
So we should have the animal on one layer the Skeleton on another and the fleshy human that we discard it somewhere else.
The next step is to repurpose the skeleton so that it matches the animal that we want.
To do that we're gone to enable only the animal layer and the skeleton layer.
Next we're going to start setting things up with Avastar.
First make sure that the experimental tab is selected in the user interface menu
Then make sure that you are in the skin and weight mode.
The next thing we need to change is how many bones are visible and what we're going to work with since this is an animal we don't need to have as many bones as we do with humans however that's just my personal opinion you can add in as many bones as you want I just find that animals are simpler and don't need Fit weight bones or all the hand bones a simple custom Classic will work.
So in the rig display you're going to scroll down to the deformed bone groups and disable the hands box, and you want to turn on the tail box.
You should have and SL Base, Face and Tail selected.
Before we end the prep stage you're going to have to enable the X modifier for the bone moving feature that we're about to use.
You don't have to use it but it'll save you a lot of Sanity being able to head know that the bones are mirrored on side rather than having to do like side a and side B.
On your blender cab you should see options it's right under the tool Tab and above the grease tab. Select your skeleton which would be in pose mode and change it to edit mode.
Check the box that says X-axis mirror.
And that's it for a preparation.
From here on out will be working in edit mode.
Select your skeleton and go into edit mode
This is where we begin changing the skeleton from a human skeleton into an animal one.
Select the skeleton and press s to scale it down to your animals size.
I want you to imagine a person inside of an animal suit on all fours pretending to walk around as a dog.
That's the skeleton shape you're going to try to recreate.
Press A and select the whole skeleton
Then press R-X- 90 and the skeleton should be facing down.
From here your to grab the bones and try to position them in that pulls I told you about earlier a human person on all fours inside of the animal's body.
I can't really explain to you how to set up your head because I don't know what kind of rabbit that you have or the one you want to rig but you know I'm pretty sure you got the gist of it once you see it like oh okay so I'm going to put the eye bones where the eye is and stretch the ear bones up with a ear should be.
Once your skeleton is all set into place you have to save the configuration so in edit mode you want to go to the avastar tab.
Making sure that you're still inside of edit mode scroll down to see posing
Then select store joint edits
Your bones will snap back to the default position which is okay this is expected you want to press the little box that says snap control to rig.
And your skeleton will be saved.
From there it's a matter of just rigging your new Avatar to your custom skeleton.
You can move around any bones and if you find that some bones are in a way you can just hide them by clicking the bone and pressing H.
So yeah once it's red you can import it in and you'll have an animal avatar.
(make sure when you're importing it that you have skin weights activated as well as the joint position)
However it won't work straight out of the gate.
You'll have to create and export a standing pose for your avatar.
Because when you import it in Second Life will try to register it as a human so we'll be kind of funky.
So you need to make a proper standing Avatar so I can stop and basically creating an Ao fo it because it won't work for the standard stuff.
It's also worth noting that sometimes when you make a custom skeleton that second life wants to be a butt and will deform your animal quite badly.
Is it happened there a few steps you can take to fix this.
1: The first one is going into avastar and exporting your shape XML.
Then importing net XML and the shape section of Second Life.
If that doesn't work then you may need to clean the skeleton because it's a small bug that's been going on recently.
And you do that inside the avastar menu in the tool section
Impressing cleanup rig now this make your bones break like they'll lose their hierarchy, and will rotate independently.
To fix that error you go into the Avastar Tab and select pools in animate select the skeleton go into edit mode,a mode press a, then in the blender bone menu the one next to the material menu
You going to scroll down until you see the relations tab,
Click the connected button and then press it again.
So you want to uncheck it then check it.
As well as the inherit rotation button sometimes that works too.
They should fix that error.
Okay so now you got your avatar all rigged and stuff you are now ready to switch it over into animesh if that's what you want.
But from there you're kind of on your own I haven't played with animesh much.
Here's some videos I made when I was making animals I recently did a dog and a spider but it was mostly for entertainment the dog so it's really long for no reason, spider is a lot shorter in right to the point.