If you feel the brambles are holding the soil in place, best to do this job well after the rainy season.

Mulch will not prevent growth or regrowth of brambles unless it is impractically thick, new plants would eventually appear at perimeter of mulched area.

And chopped off crowns will come right up through any mulch.

They need to be pulled up or sprayed with blackberry killer. The tips are already rooting in new areas.

I would also wait until the rainy season is over, then dig out as much as possible by the roots. Then you could apply netting and grass seed, or plant ground cover plants and mulch around them. When any shoots from the blackberries appear, hit then right away with Roundup, being careful not to get it onto good plants.

Glyphosate label - if still the same as in the past - says to spray blackberries in autumn so the crowns will be killed. Small shoots early in the season probably aren't going to take up enough of the chemical to kill the entire plant.

Crowns gone after when the soil has become well moistened - such as in spring - will be easier to get out.