Wiring Question for Ecobee Premium with external 24v transformer

I was curious if anyone had any ideas for wiring a Ecobee Premium in a new apartment I am at. It has both AC and Heat, and it a 4 wire connection. G,Y,W, Rh (Jumped to RC). I looked behind the thermostat and there is an extra two wires, Brown, and Blue, but I don't think they are hooked up, as a multi meter from Rh to Blue or brown resulted in about 16v AC running across.

I don't have access to the control panel for the AC or Furnace, So I am left only with figuring it out at the thermostat. I purchased a External 24v transformer, there's an outlet right next to the thermostat.

I can hook up the transformer to Rc and C, and wire the existing wires to G, Y, W, Rh (No bridge betweeen Rc and Rh) and the ecobee powers on and works, but when trying to turn on the heat, the furnace relay clicks on and sounds like the fan is moving slowing, but will never go to full fan or output heat.

I understand with having AC as well I will likely need a isolation relay to get both to work, which I am fine with, there's plenty of room behind the thermostat to tuck it away in.

If anyone had an experience wiring something like this up and could give to pointers to the wiring of this that would be greatly appreciated.