I am trying to plumb my 1/8" BSPT espresso machine through a water filter with both 3/8" FNPT inlet and outlet into the water outlet behind my refrigerator. So far, I have a 1/8" BSPT to 3/8" NPT adapter, and I have ordered a stainless 3/8" M to 3/8" M coupler/nipple. I believe this should cover everything from the filter outlet to the machine. You can see this part (minus the nipple/coupler) on the right hand side of the filter.
The problem I have now is trying to find a suitable connection from the water outlet. I believe the outlet has a 1/8" fitting, but am not sure what standard this is. As seen in the above picture in the bottom left, there was some kind of compression fitting on it. The outlet measures ~.44" OD as measured from the wall, and the compression nut measures ~.39" ID as seen in the picture above. The outlet itself comes off a shutoff valve. The shutoff valve body has a 1/2" M brass pipe fitting that threads directly into the wall.
So my question is what is the best way to connect the water filter inlet to the wall water outlet. I am willing to replace the shutoff valve with something different if it makes sense. This is all going to be used for potable water, so something with no brass would be even better. I tried going to the hardware store, and they didn't have any solution as the water filter inlet is 3/8" FNPT and all their shutoff valves used compression fittings which were a different size.