What you are proposing wil serve no practicle purpase and it is highly illegal.

Here in Ontario a grey water bed is referred to as a Class 2 bed. If you had one prior to 1986 you are able to use it if not we can't create new beds.

I wouldn't be concerned with freezing in winter if you get the drain line to the below the frost level within the first 4 feet of run. Then the standard 1/4 " per foot drain to the new bed. The drain for a washer is under pressure and unless the line is plugged water won't freeze in the line. The bed itself is another issue, it is not just a hole in the ground. Another thing to consider is the drain line diameter, lint from the washer will plug a 2 inch drain over time.

After all is considered especially the environmental impact, paying to haul waste from a septic bed is not the worst thing one can do.

Another thing is if you are not allowed to create a grey water bed, you are going to reverse everything if you decide to sell.

I would not do it.

I must add you are at risk contaminating your well with off run from the new greywater bed.