Hiya,
I tried this last year. I found that when the Jerusalem Artichokes were getting going, early in the season, I had to go around and manually pull out the bracken stalks - which were coming up much faster.
Once the Jerusalem artichokes were underway, however, they did indeed seem to suppress bracken growth to some degree, as well as other fast-growing aggressive plants like comfrey and thistle. I had planted them somewhat sparsely and the only maintenance I did was maybe five or ten minutes of pulling out weeds around them (not even all the weeds), which I did only about twice over the whole summer.
I'm hoping the patch will come back thicker this year and continue to do this job even better. The other side of the small field had no Jerusalem Artichokes and suffered from a mass of bracken and bramble, which I cleared by hand and used as mulch on other things.
I will note that the Jerusalem Artichokes in this patch did not do nearly as well as the ones grown in my other land, where they don't have to compete with anything except grass, clover, etc. But they still grew tall and yielded a good crop, with no fertiliser or anything. Literally just chucked under bits of sod lifted with a spade - not even proper holes.
I think a very thick patch of JA, grown year after year (maybe with mulch in the first few years to help establish it) would probably be very effective and healthy large tubers left in the ground would help it get a head start on the bracken.
Even now, the ground where the Jerusalem Artichokes grew has a lot less weeds and shorter grass than the other side. It's interesting.
Bracken also makes great mulch for potatoes, I'm finding, if you have lots of it to spare. No need to
compost it.