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We use nuclear magnetic resonance (NMR), high-resolution x-ray and neutron scattering to study structural and magnetic phase transitions in phosphorus-doped BaFe$_2$(As$_{1-x}$P$_x$)$_2$. Previous transport, NMR, specific heat, and magnetic penetration depth measurements have provided compelling evidence for the presence of a quantum critical point (QCP) near optimal superconductivity at $x=0.3$. However, we show that the tetragonal-to-orthorhombic structural ($T_s$) and paramagnetic to antiferromagnetic (AF, $T_N$) transitions in BaFe$_2$(As$_{1-x}$P$_x$)$_2$ are always coupled and approach to $T_N\approx T_s \ge T_c$ ($\approx 29$ K) for $x=0.29$ before vanishing abruptly for $x\ge 0.3$. These results suggest that AF order in BaFe$_2$(As$_{1-x}$P$_x$)$_2$ disappears in a weakly first order fashion near optimal superconductivity, much like the electron-doped iron pnictides with an avoided QCP.