When you say "... redone by a landscape masonry firm." ... you mean relaid on a granular base? (...not mortared in?) The walk needs to be completely pulled up and reset. I would be willing to bet that there is no proper base. Having a new base will help a lot in keeping the weeds away as they won't prefer it to soil. But some will come and the best/easiest way to dispense with them is a periodic spray of herbicide. When you say "permeable brick pavers," what exactly are you speaking of? Some people might call what you already have as that. If you mean the kind of pavers that grass is supposed to grow in, then you will probably have a time with with weeds there. It is a rare case when those pavers look good. Most of the time they look like a muddy area with bricks embedded. It would be a great improvement if when you do the walk, you widen it. It's too narrow. 4' would be better.

My office is in Fairfax County; I've specified a few permeable pavements for clients in FFX so they can build sport courts, etc. If you mean a true permeable paver system like Techo-Bloc's Permea (link below), weeds won't be as big an issue if it's properly installed. The pavers are installed on a bed of 3/4" clean stone, then a layer of filter fabric, and then a bedding layer of 3/8" chip gravel which also goes between the pavers. Nothing's coming up from below. Yes, weed seeds will occasionally blow in and land in the joints, but as long as you keep up with it it shouldn't be too terribly awful.

I 100% agree with Yard that if you're redoing the walk, make it wider. I like to walk next to my wife when we're walking up to a friend's front door. 5' is my personal minimum for a front walk.

Here is a link that might be useful: Techo Permea