Something's fishy... First of all, that bypass job was done by a redneck plumbing school dropout (sorry if I offended any friends who didn't make the cut in plumbing collige...). You could run it for a while with hoses attached, but my guess is that you'll find (at the least convenient time) that your core is leaking. The core can leak for a while, then seal up, and then start leaking again - it's the weakest part of your cooling system. If you are going to test this out with the core in the loop, you should take your passenger side carpets and floor mats out. It is much easier to keep them from getting that sickly sweek stink in them than trying to get that smell out of them. It will also be easier for you to see the fluid drippin down on your floorboards.
All in all, if that were my car, I'd be heading down the path of ordering a new heater core and installing it. It's not a horrible job, but it's not much fun, either. But, it's less work to just replace it now than to have to clean up that nasty smelling antifreeze from the passenger footwell and THEN replace the core. Most cars of that vintage, if they haven't had regular antifreeze changes (especially if the owner ran the green stuff in them the whole time) are due for a new heater core, and possibly other coolant related items. Just my $0.02.