Whilst fully relativistic lattice calculations working at the physical $b$ quark mass are now possible, their widespread use remains unfeasible. An alternative is the so called heavy-HISQ formalism, which uses heavy masses, approaching that of the $b$, to provide precise calculations in $b$ physics. Here we discuss HPQCD's improved scalar, vector and tensor form factors for $B\to K$ semileptonic decays. Working with masses close to the physical $b$ and including three ensembles with physical light quarks, we cover the full physical range of momentum transfer with good precision. We compare Standard Model observables, obtained using our form factors, to experimental measurements for the rare flavour changing neutral current processes $B\to K\ell^+\ell^-$ and $B\to K\nu\bar{\nu}$, and discuss the significance of the tensions that arise.