Weaker? Not exactly.
GPU, CPU and amount of RAM should be way higher than PS4.
Brandwidth is more related to the resolution you want to render.
PS4 run almost all games in native 1080p and so the bandwidth is enough.
For games in lower resolution like 720p, 800p, 900p, etc you just need around half of it (Steam Deck come to mind... and that is the biggest reason Steam Deck falls below PS4 if you set it to 1080p... or Xbox One
).
So with that memory bandwidth (120GB/s) I believe Switch Pro is not aiming to native 1080p but lower res screen like 720p... and for a portable it is really fine.
Of course it will upscale to 1080p or 4k (?) in docked mode.
Here the DLSS comes to mind too.
Base resolution at 720p or lower upscaling via DLSS to higher resolution... it doesn't need memory bandwidth for 1080p resolutions.
It is actually a killer setup for a handheld screen... and DLSS can make up the difference for a docked screen.