Yes, you can get cheap 3D BDPs. For example, the excellent (EDIT: 2010 BluRay product of the year) Sony BDP-S570 can be had for £165 from RGBDirect, or £175 from Dixons.
It's possible that your HDMI cable might carry a 3D signal. Possible, not definite. It definitely won't have an audio return channel or ethernet, but it might have enough bandwidth for 3D. Really depends on manufacturer, model and length of cable and how much over 1.3a/b/c spec they went...