I asked because depending on how intricate the design is, or how delicate the finish play's a factor in which brushes to recommend as all vary in size, aggressiveness etc.
Below are my Weekend Warrior recommendation's, not a Pro "Detailer"
Tyre (In2Detailing)
I find a dedicated Tyre Brush (With firmer bristles) are ideal for deep cleaning tyres, especially during the winter period and are inexpensive.
Wheel Barrel Brush (Autobrite Direct/Amazon)
Autobrite 's Large Flexi Wheel Brush - This works wonder on the inner barrels, extremely light weight when loaded up with suds due to having Nylon Bristles.
Alternatively you could go down the Microfiber which are much softer, hold plenty more solution. £10 with Free Delivery (First Order you can get a 15% Discount also)
(Turtle Wax Directly)
Sometime's it comes down to preference as there are endless choices out there, none necessarily being the wrong choice.
Wheel Face
Considering your design, as it's not extremely intricate and you mention "Efficiency" perhaps a larger style wheel face brush with soft bristles would be a good option rather than a slim detailing brush?
(Clean Your Car)
Alternatively