I'm going to have a go at making a double fleece indoor coat for my Great Dane and I'd like to bind it with something contrasting.

This may sound stupid but can a normal sewing machine cope with 2 layers of fleece and the binding?
Do I need a special needle?
Also does binding come in different thicknesses as the one I have seems quite thick?

I'm very rusty at this but I'm hoping that as I used to overlock and bartack that I'll get the hang of it quickly, especially as I'm hoping to do loads of home furnishing myself for the new house.

Any help appreciated.