I just skimmed through all the replies so may have missed something.
What I would do is turn off the floor heat and turn up heat in house and wait a few minutes for water temp in return line to stabilize. Then open the valve in the main line and see if there is a large increase in the return line temperature.
If there is, then the plate is way too restrictive. You may be able to get it to work better by partially closing the valve as mentioned above but you probably won't get enough BTUs for the floor and even partially closed will still be slowing down the flow.
If it would be a big pain to re-plumb for a 5x12 heat exchanger then I would just get a 3x7 50 plate or larger.
How many pex elbows are in the line? I would remove the ones shown in the picture and just use plastic bend supports instead on a new single piece of pex.
Another option that may be required is to put a pump between the 1st tee and the plate heat exchanger and run at the same time as the floor pump. If you do that, you can probably leave the rest of it the way it is.