No the bed thickness has no impact on strength,,the strength is in the mix, obvious the thicker the bed then use something suitable, , suppose it was 4 inch , then use concrete,its just a matter of mix consistency , to keep it in the joint and not running out,you might have to shutter the joint if you use a wetter mix to stop it running out.
Or cut the joint ,bed, back and point it after it has set. to get a good finish.