@Sl33pydog Almost, i edited your pic. One winter setup when the air outside is lower than inside and using the radiator. And one summer-setup where the peltiers work. I also changed the peltier-loop a bit, as it´s separate from the other loop. Two pumps in the tank, 2 hoses out, 2 in. One pair for the tube, one pair to the peltiers.
A radiator at the same time as a peltier, you will actually heat up the water as a rad. only can cool to ambient temps.
If the peltier pulls down the tank to 18c, and your ambient is 22c, you will be “heating” the 18c water with 22c air 
Take that into concideration depending on your temps.
I have a W1209 thermostat controller with a temp-probe, it turns on at 18c and turns off at 14c, well, never turns off on heavy work 
At 24c ambient i have 16-17c for 20-30min with six 60w peltiers.
I´m thinking of watercooling the hot-side of the peltiers too, just to press some extra cooling out of them 
My setup is not portable, not actually 100% closed either as i have a small vent for pressure differences.
25 litres of water so it´s very heavy 
On your idea you do not need one, just make sure to bleed it from air and make sure warm water can expand some inside the loop.
Computer cooling pumps works good, i used both D5 and Laing DDC pumps earlier. Now i use 12v bilge pumps for boats rated 14L/min as i can have them inside the tank. (25L plastic tank i seal when filled)
Having pumps on reastats is a good idea, you can tune it to whatever you want.
I use a arduino with a safety system on mine, it controls water temp, water flow, mirror temps (to notice me when to clean them) and in the winter it controls the fan speed on the radiator.
Just moved my machine to another room so my setup is in pieces, but i can shoot some pics on different parts to show what i use.
Edit: i forgot one pump on the summer setup
Should have one per loop 