The biggest benefit would have been if it supported Vulkan, but that wasn't added until the version after － Crimson 16.3. In fact, this is likely why the newest beta version is 16.2.1 as your GPU uses the older TeraScale architecture which does not support Vulkan at all (not even on Linux).
Otherwise you're likely better off just using whatever drivers HP provides on their website for that specific laptop considering that it sounds like your laptop has switchable graphics.
Amusingly, the Intel integrated graphics can actually support Vulkan on Linux, but that'd probably require running at like 1x internal resolution (maybe 2x) for any chance of getting better performance, but that's without regard for ubershaders. Nevertheless this would almost certainly require using dual channel RAM, and it isn't clear if you have 2x4GB or 1x8GB.
Otherwise you're likely better off just using whatever drivers HP provides on their website for that specific laptop considering that it sounds like your laptop has switchable graphics.
Amusingly, the Intel integrated graphics can actually support Vulkan on Linux, but that'd probably require running at like 1x internal resolution (maybe 2x) for any chance of getting better performance, but that's without regard for ubershaders. Nevertheless this would almost certainly require using dual channel RAM, and it isn't clear if you have 2x4GB or 1x8GB.
Dolphin 5.0 CPU benchmark
CPU: Pentium G3258 @ 4.5GHz 1.24v
GPU: Intel integrated
RAM: 4x4GB Corsair Vengence @ DDR3-1600
OS: Linux Mint of some variety + [VM] Win7 SP1 x64
CPU: Pentium G3258 @ 4.5GHz 1.24v
GPU: Intel integrated
RAM: 4x4GB Corsair Vengence @ DDR3-1600
OS: Linux Mint of some variety + [VM] Win7 SP1 x64