I had a problem once, with a used back, on a fairly new body (purchased new by me, used only a couple of years), a long time ago. It turned out to be an issue that the interlock pin in the back would sometime push a bit too far into the body, causing the reflex mirror to not be able to swing up fully, so the rotary mechanism at the back of the lens would not rotate to permit the shutter to fire! Very intermittant issue, which I finally linked to one of my multiple backs (the only one I had purchased used, at the time of the problem).
Bronica (distributor GMI Photographic, in the USA before Tamron purchase of Bronica) told me to send the body and the back to them for adjustment. They adjusted the pin in the back to fix the issue.
In your case, you experience the problem with multiple backs, so your issue is not identical to mine. But I surmise that an adjustment to the mechanicals within the body needs to be adjusted, which is why changing backs does not affect the issue for you. I fear that, as Bronica has not been sold new for some time, that the necessary expertise to do the adjustment may not be found. I view of that lack of expertise, and the incredibly low prices at which Bronica gear can be procured -- only about 10% of the price when new ! -- I would simply seek an ETRSi body to replace your ETR, and perhaps keep the ETR only 'for parts'.