Mowing is good for controlling brush but will create more sun for Blackberries.
It would be helpful to know your location. Blackberries in Florida or Georgia are a much different case from Blackberries in the Pacific Northwest.
I favor the herbicide Crossbow or Rural King's equivalent 'Crossroad'. Crossbow/Crossroad is 34.4% 2,4-D enhanced with 16.5% Triclopyr herbicide, with Blackberries specifically as a target. 2,4-D in general is for Broadleaf Weeds, which is Blackberries.
All 2,4-D herbicides are most effective when the target weeds are growing vigorously. This means Spring or Fall in most areas. The soil should be moist. Ideally, apply herbicide 24 hours before a soaking rain. Soaking rain carries some Crossbow from the leaves down to the roots.
Using Crossbow/Crossroad over ungrazed pasture in a one gallon sprayer in North Florida I have had 50% Blackberry kill after one application and almost 100% kill after a second application a month later. Pasture grass yellows a bit but recovers.
Read the herbicide directions, FOLLOW THE DIRECTIONS. Do not create home brew herbicide mixtures.
Bathe and wash clothes as soon as practical after spraying.
Crossbow is a non-selective post-emergent herbicide that targets woody plants and brush such as blackberries and poison oak, as well as annual and perennial broadleaves, while leaving grasses unharmed. Crossbow achieves excellent control of woody plants and brush and can be applied by all types of sprayers, from hand-held models to tractor-mounted spray rigs. Except for lactating dairy animals there are no grazing restrictions following the application of Crossbow Herbicide. Crossbow is rainfast within two hours after application.
This is Not to be applied to home lawns.
Note: Due to New California State restrictions, this Can Not be Sold in California.