Saying that a quantity can be measured with arbitrary precision means that you can measure it with any non-zero level of precision. So if you pick some particular non-zero number $\Delta E$ there is some measurement that could measure the energy of whatever process you're interested in with that precision. The fact that you have to measure energy with time dependent processes implies that $\Delta E$ must be non-zero, but that doesn't imply that measuring the energy with arbitrary precision is impossible.
Real measurements are interactions that produce records. Those interactions suppress interference: a process called decoherence:
https://arxiv.org/abs/quant-ph/0306072
Decoherence produces states that are narrowly peaked in position, momentum and other quantities such as energy on the scales of everyday life
https://arxiv.org/abs/0903.1802
https://arxiv.org/abs/1111.2189
Depending on what kind of interaction you choose you can make the widths of the states in terms of energy or momentum or whatever narrower or wider, but you can't break the uncertainty principle. And the limits imposed by the uncertainty principle are very small on the scale of everyday life. For example, if you have a particle with $\Delta x = 10^{-7}m$, the lower bound on the uncertainty in momentum is $\Delta p > 10^{-27}kgms^{-1}$.
Nor can you make the uncertainty in position zero since real measurement devices are governed by differential equations of motion whose solutions are differentiable and so aren't infinitely narrowly peaked in position.