What makes sense is to do both jobs on the same shop visit, but not necessarily have them done in parallel. The reason is practical: if they're going to have the car up in the air checking the suspension anyway, they might as well swap out the brake pads without taking the vehicle down again. That saves you shop time and mechanics often knock a bit off labor when jobs overlap like that.
Now here's the trick that people don't mention: before you commit to anything, ask the shop to spell out exactly which suspension components are damaged. It's not the same thing to just replace shocks versus swapping out bushings, control arms, or sway bars. Worn suspension is a really broad term. Some shops will tell you that you need the whole job done when maybe only a few parts actually need urgent attention. I always get a written diagnostic with details before I say yes to anything.
In your case, that dip when you brake is a serious warning sign you shouldn't ignore - the front suspension is probably compromised and that's affecting how your car distributes weight during braking. But before you spend the money, get a second opinion from another shop, even just to confirm. A lot of times the same problem has cheaper solutions if someone else takes a look. Once you make the call, yeah, do it all at once.