How much straw??

I haven't got outside bunnies but I'm guesing a good thick layer. Better safe than sorry I don't think you could put too much in.

I use a depth deep enough to bury my arm, so it's nice and soft for little bunny bums

My sister puts a lot in. If there's too much the buns will pull it out, then you know for next time.

I'd only put your snugglesafe in when it's cold.

Might I suggest using hay instead of straw, as it does not have such sharp ends which can cause eye injuries.
I use the left-over uneaten hay at the end of the day to add to the bedding pile, and older used hay from there to put in the loo tray.
When the loo needs cleaning, the lot goes into the garden recycle bin.
Bales of hay are cheapest to buy at around £3, and last ages for a bun.
They then have a constant supply to nibble, and it gets the most use possible.

I do the same with the left-over hay and use it as bedding. I do use the very soft straw in winter as well though as it has higher insulation properties than hay. The best I've found is the Russell Rabbit extra soft straw. I've a snugglesafe but as I've two bonded rabbits outdoors who use each other as hot water bottles, I now use it for sick bunnies who need to be kept warmer than usual.

I use about three inches of straw and then a good fat layer of fresh hay on top. This is their main hay supply so that's why it's fresh. If you have a good separate supply of hay elsewhere, you need only put in straw or if you wanted to, put a thin layer of hay on top of the straw, but remember, most bunnies burrow, so if you are trying to aviod eye injury, the hay on top is pointless. My Milly burrows under the hay and straw and she's fine. (fingers crossed) Straw is not usually that sharp anyway (good quality straw), and bunnies know when to shut their eyes and to get out of danger. They're not stupid.

ETA: I only use a Snugglesafe when it's like below 0C.

It doesn't matter how much I put in, they drag it all out and make a new space anyway. In the coldest parts of the winter I was loading it with both hay and straw, but they would still pull it out and dump it in their run - very particular about house cleaning are my buns!
I guess that the two of them together keep them warm enough.

To echo what was said above, just try and see what happens.