Rheinisch qualified for the final in 10th place and only made it when favourite Peter Kauzer of Slovakia, the last man down the course, was given four penalties.
He then went clear again as first man in the final and was holding on to bronze until the last canoeist Benjamin Boukepti (Togo) beat his time.
On coming fourth:
“It’s heartbreaking that I’m so close to a medal.”
“One error is what cost me a medal, but I was really fortunate to come this far.I can’t complain, I’ve been riding my luck to get just through each heat to the final.