I have the same Sony camera and spent a while working out whether any good mounting options existed before buying one instead of a GoPro.
As it turns out, you can easily use all of the GoPro mounts by buying a Veho Universal Tripod Mount
To use it, you need to undo the bolt through the middle of the Veho mount separating it into a top and bottom section. The top section then screws into the standard tripod mount on the Sony’s waterproof case, and the plastic blades part bolts into the GoPro mounts.
There is a good video somewhere on YouTube that goes through this process but I can’t find it at the moment.
I have used this successfully with both a GoPro Chesty mount and rear facing from the seatpost using the GoPro handlebar mount kit.
I was a little concerned about how far it would stick out when chest mounted, but it’s never been a problem when riding (I mount it upside down and flip the video afterwards on a computer) and I doubt it actually sticks out much more than the GoPro.
I’ll try and post some pictures later when I have access to my camera and mounts.
Hope that helps.