Can you bypass the sleeper heater core?

It's not a bad design, it's just wrong. I would just fix it, all you need is 2 of these in the correct size:
https://static.grainger.com/rp/s/is/image/Grainger/6AFH7_AS01?$zmmain$

Your heater hose is probably 3/4" ID (maybe 5/8") and the tee looks like 1/2" NPT.

 

My crystal ball says they are 1/2 male pipe to hose fittings needed. Those fittings are usually smooth with a hump at the end. Barbs have multiple humps which would work also but not the norm in heater hoses.
Do away with two fittings under the hoses and use one. The hoses look to be 5/8's. If a 5/8 open end wrench will fit over the tube ,a little loose, where the top of the hose connects they are 5/8. If it won't they are 3/4.
Over sized clamps have a longer area at the worm gear. This can cause them to be unable to conform to the smaller circle and leak, as mentioned.
The blue hose is a superior silicon hose but they are softer and require a special clamp. They have an inner band inside the clamp that prevents pushing part of the hose through the slots in the clamp. This happens over time loosening the clamp and then leak.
Who ever thought so much could be said about a couple of heater hoses?
By the way it is marked on the ball valve, 1/2. Thanks Grape! I see that took me 30 minutes.

 

Ditto on what boxcar said about the blue (silicon) hose. They take their own kind of clamp. The best is the ones that you don't tighten but use pliers or the fancy tool to squeeze the ends and then release them once they are in place. Them seem to do the best just due to heat (expansion contraction thing)

 

I only use dope in my shop, only because it's easier to not allow teflon tape then it is to get people to use it correctly. It only takes 2 or so wraps and keep it on the threads. Sounds simple enough, but it isn't. I have no idea how people manage to get it inside the fittings, but on tear downs, I have found chunks of teflon tape inside things. I recently found a brake problem with a piece of teflon tape in a spring brake check valve (the one that holds air in the trailer tanks). When the brake were set, all the air dumped out of the trailer through the MV3 becasue the check valve couldn't seat.

With all that said, either works just fine when used correctly. If I were you, I'd get enough silicone hose to do all of your heater hoses and spend an afternoon to replace them all correctly and you be trouble free for years. That black hose in the picture is looking pretty shabby. It'll save you money in the long run.

 

That's a lot better than clamping onto a pipe nipple. Personally, I would have found pipe nipples to screw right into those tee's and used longer hoses to eliminate 6 fittings. The simpler the better, more fittings only means more potential leaks, but what you have there should work just fine as long as all those fitting are sealed up good.

 

I can't find those two valves, I have a Paccar motor 386 pete

 

Those four clamps are way too tight. Something isn't right for all 4 of those connections. You need to put something on all 4 with teeth. Too hard to convert to English and describe for me.